Systems and methods for energy efficient electrolysis cells

ABSTRACT

Disclosed herein are systems and methods for obtaining efficient aluminum smelters. More specifically disclosed herein is a method comprising: applying an alternating current (AC) comprising an oscillatory current waveform to an electrolytic cell comprising an electrolyte for a first predetermined time, wherein waveform comprises an amplitude, frequency and/or phase that are predetermined to stabilize the electrolytic cell such that substantially no change in a current oscillation is observed in the electrolyte during electrolysis. Also disclosed herein is a system comprising an electrolytic cell, direct current and alternating current sources. The disclosed electrolytic cell exhibits substantially no change in oscillations present in the molten salt electrolyte over a predetermined period of time when the AC is provided to the electrolytic cell.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/972,286, filed Feb. 10, 2020, the content of which is incorporatedherein by reference in its entirety.

STATEMENT ACKNOWLEDGING GOVERNMENT SUPPORT

This invention was made with government support under Grant No.CBET-1552182 awarded by the National Science Foundation. The governmenthas certain rights in the invention.

BACKGROUND

Currently, aluminum is produced by running large, steady (DC) electricalcurrent through pools of molten salts atop liquid aluminum, a processthat consumes about 3% of electricity worldwide (˜10¹¹ kWh/year) (P. A.DAVIDSON and R. I. LINDSAY. Stability of interfacial waves in aluminumreduction cells. Journal of Fluid Mechanics, 362:273-295, 1998). About40% of that energy is wasted as heat in the salt and could be saved ifthe salt layer were thinner. Unfortunately, thinning the salt allows aresonant instability in which surface waves grow uncontrollably untilthe smelter is shut down.

Thus, there is a need for systems and methods for producing aluminumthat allows reducing the thickness of the salt layer, and thus,increasing the efficiency of the energy consumption without a compromisein an aluminum smelter stability. These needs and other needs are atleast partially satisfied by the present disclosure.

SUMMARY

The present invention is directed to a method comprising applying analternating current (AC) comprising an oscillatory current waveform toan electrolytic cell comprising an electrolyte for a first predeterminedtime, wherein waveform comprises an amplitude, frequency and/or phasethat are predetermined to stabilize the electrolytic cell such thatsubstantially no change in a current oscillation is observed in theelectrolyte during electrolysis.

In still further aspects, also disclosed herein is a method comprising:a) providing a first data to a computational processor, wherein thefirst data comprises at least one of one or more of geometric parametersof an electrolytic cell, a cathode-to-anode-distance of the electrolyticcell, a value of a direct current; an amplitude of a direct current, athickness of a metal layer, material properties of a metal, materialproperties of an electrolyte, material properties of a cathode, materialproperties of an anode, or any combination thereof; b) analyzing thefirst data by the computational processor to provide a second datacomprising parameters of an alternating current (AC) wherein theparameters comprise one or more of a first amplitude, a first frequency,and/or a first phase of an oscillatory current form of the AC; and c)applying the AC having one or more parameters present in the second datato the electrolytic cell to stabilize the electrolytic cell. In yetfurther aspects, the method is further comprises d) collecting a thirddata from the electrolytic cell and transferring the third data to thecomputational processor to analyze the performance of the electrolyticcell; e) analyzing the third data by the computational processor toprovide a fourth data comprising parameters of the alternating current(AC) wherein the parameters comprise one or more of a second amplitude,a second frequency, and/or a second phase of an oscillatory current formof the AC; and f) applying the AC having one or more parameters presentin the fourth data to the electrolytic cell.

In yet further aspects disclose is also a method for increasing energyefficiency in an electrolytic cell comprising: applying an alternatingcurrent (AC) comprising an oscillatory current waveform to theelectrolytic cell comprising an electrolyte for a first predeterminedtime, wherein waveform comprises an amplitude, frequency and/or phasethat are predetermined to stabilize the electrolytic cell such thatsubstantially no change in a current oscillation is observed in theelectrolyte during electrolysis; and wherein the energy efficiency isincreased by at least about 5% when compared to a substantiallyidentical reference electrolytic cell in the absence of applying an AC.

In yet further aspects, disclosed is a system comprising: a) anelectrolytic cell comprising: i) an anode; iii) a cathode; and iii) anelectrolyte having a predetermined thickness; b) a direct current sourcethat is in electrical communication with the electrolytic cell and isconfigured to provide a direct current (DC) having a predeterminedamplitude and to initiate an electrolysis reaction in the electrolyticcell; c) a device comprising an alternating current source (AC); whereinthe device is in electrical communication with the electrolytic cell andis configured to provide an alternating current (AC) to the electrolyticcell, wherein the AC comprises an oscillatory current waveform definedby a predetermined amplitude, frequency, and/or phase; and wherein thedevice is in feedback loop communication with the electrolytic cell; andwherein the electrolytic cell exhibits substantially no change inoscillations present in the molten salt electrolyte over a predeterminedperiod of time when the AC is provided to the electrolytic cell.

In still further aspects, the disclosed herein electrolytic cell is analuminum electrolysis cell. In yet some aspects, the molten saltelectrolyte comprises cryolite. In yet further aspects, thepredetermined thickness of the molten electrolyte can be equal to orless than about 4.5 cm.

Additional aspects of the disclosure will be set forth, in part, in thedetailed description, figures, and claims which follow, and in part willbe derived from the detailed description, or can be learned by practiceof the invention. It is understood that both the foregoing generaldescription and the following detailed description are exemplary andexplanatory only and are not restrictive of the invention as disclosed.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 a schematic of Hall-Heroult cell (H-H cell) (according to P. A.DAVIDSON et al., “Stability of interfacial waves in aluminium reductioncells.” Journal of Fluid Mechanics, 362:273-295, 1998).

FIG. 2 depicts an exemplary schematic diagram of an exemplary system inone aspect.

FIG. 3 depicts an exemplary schematic diagram of an exemplary system inone aspect.

FIG. 4 depicts a diagram of an exemplary mechanical model that isanalogous to the sloshing instability in aluminium cells (according toP. A. DAVIDSON et al., “Stability of interfacial waves in aluminiumreduction cells.” Journal of Fluid Mechanics, 362:273-295, 1998).

FIGS. 5A-5C depict exemplary models in one embodiment: FIG. 5A—a modelof the unperturbed state; FIG. 5B—a model under small rotation aboutx-axis; and FIG. 5C—a model under small rotation about the y-axis.

FIGS. 6A-6B depict exemplary models in one embodiment: FIG. 6A—depictsLorentz and gravitational forces in x-y view; FIG. 6B—depicts Lorentzand gravitational forces in the z-x view.

FIG. 7 depicts an exemplary model in one embodiment.

FIG. 8 depicts an exemplary model in one embodiment.

FIG. 9 depicts an exemplary model in one embodiment.

FIGS. 10A-10D depict oscillations of an exemplary aluminum pot model inone aspect: FIG. 10A—shows oscillations when the direct current isapplied for 1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, anda=1.100204; FIG. 10B—shows oscillations when the direct current isapplied for 1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, anda=1.100206; FIG. 10C—shows oscillations when the direct current isapplied for 1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, anda=1.100207; FIG. 10D—shows oscillations when the direct current isapplied for 1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, anda=1.100208.

FIGS. 11A-11C depict oscillations of an exemplary aluminum pot model inone aspect: FIG. 11A—shows oscillations when the direct current isapplied for 60 s, with ω_(x)=1.5302 Hz and ω_(y)=0.3756 Hz, anda=1.100205; FIG. 11B—shows oscillations when the direct current isapplied for 2×10⁴ s, ω_(x)=1.5302 Hz and ω_(y)=0.3756 Hz, anda=1.100205; FIG. 11C—shows oscillations when the direct current isapplied for 2×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.3756 Hz, anda=1.100207.

FIGS. 12A-12I depict oscillations of an exemplary aluminium pot model inone aspect: FIG. 12A—shows oscillations when the direct current (DC) isapplied for 1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, anda=1.1003 and alternating current (AC) with ω_(b)=6.2832 rad/s and ϕ=0rad, and β=0.1; FIG. 12B—DC current for 1×10⁴ s, with ω_(x)=1.5302 Hzand ω_(y)=0.37559 Hz, and a=1.1003 and AC current with ω_(b)=6.2832rad/s and ϕ=0 rad, and β=0.15; FIG. 12C—DC current for 1×10⁴ s, withω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, and a=1.1003 and AC current withω_(b)=6.2832 rad/s and ϕ=0 rad, and β=0.2; FIG. 12D—DC current for 1×10⁴s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, and a=1.1003 and ACcurrent with ω_(b)=6.2832 rad/s and ϕ=0 rad, and β=0.25; FIG. 12E—DCcurrent for 1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, anda=1.1003 and AC current with ω_(b)=6.2832 rad/s and ϕ=0 rad, and β=0.3;FIG. 12F—DC current for 1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559Hz, and a=1.1003 and AC current with ω_(b)=6.2832 rad/s and ϕ=0 rad, andβ=0.35; FIG. 12G—DC current for 1×10⁴ s, with ω_(x)=1.5302 Hz andω_(y)=0.37559 Hz, and a=1.1003 and AC current with ω_(b)=6.2832 rad/sand ϕ=0 rad, and β=0.4; FIG. 12H—DC current for 1×10⁴ s, withω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, and a=1.1003 and AC current withω_(b)=6.2832 rad/s and ϕ=0 rad, and β=0.45; and FIG. 12I—DC current for1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, and a=1.1003 and ACcurrent with ω_(b)=6.2832 rad/s and ϕ=0 rad, and β=0.5.

FIGS. 13A-13C depict oscillations of an exemplary aluminium pot modelhaving a stability parameter a=1.1354 in one aspect: FIG. 13A—showsoscillations when the direct current is applied (DC) for 2×10⁴ s (β=0,no AC), with ω_(x)=1.5302 Hz and ω_(y)=0.3756 Hz; FIG. 13B—DC currentfor 2×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.3756 Hz, and AC currentwith ω_(b)=π/2 rad/s and ϕ=0 rad, and β=0.23; FIG. 13C—DC current for2×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.3756 Hz, and AC current withω_(b)=π/2 rad/s and ϕ=0 rad, and β=0.24.

FIGS. 14A-14I depict oscillations of an exemplary aluminium pot model inone aspect: FIG. 14A—DC current for 1×10⁴ s, with ω_(x)=1.5302 Hz andω_(y)=0.37559 Hz, and a=1.1003 and AC current with ω_(b)=0 rad/s and ϕ=0rad, and β=0.1; FIG. 14B—DC current for 1×10⁴ s, with ω_(x)=1.5302 Hzand ω_(y)=0.37559 Hz, and a=1.1003 and AC current with ω_(b)=0.7854rad/s and ϕ=0 rad, and β=0.1; FIG. 14C—DC current for 1×10⁴ s, withω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, and a=1.1003 and AC current withω_(b)=1.5708 rad/s and ϕ=0 rad, and β=0.1; FIG. 14D—DC current for 1×10⁴s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, and a=1.1003 and ACcurrent with ω_(b)=2.3562 rad/s and ϕ=0 rad, and β=0.1; FIG. 14E—DCcurrent for 1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, anda=1.1003 and AC current with ω_(b)=3.1416 rad/s and ϕ=0 rad, and β=0.1;FIG. 14F—DC current for 1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559Hz, and a=1.1003 and AC current with ω_(b)=3.927 rad/s and ϕ=0 rad, andβ=0.1; FIG. 14G—DC current for 1×10⁴ s, with ω_(x)=1.5302 Hz andω_(y)=0.37559 Hz, and a=1.1003 and AC current with ω_(b)=4.7124 rad/sand ϕ=0 rad, and β=0.1; FIG. 14H—DC current for 1×10⁴ s, withω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, and a=1.1003 and AC current withω_(b)=5.4978 rad/s and ϕ=0 rad, and β=0.1; and FIG. 14I—DC current for1×10⁴ s, with ω_(x)=1.5302 Hz and ω_(y)=0.37559 Hz, and a=1.1003 and ACcurrent with ω_(b)=6.2832 rad/s and ϕ=0 rad, and β=0.1.

FIGS. 15A-15C depict oscillations of an exemplary aluminium pot modelhaving a stability parameter a=1.1354 in one aspect: FIG. 15A—DC currentfor 2×10⁴ s (β=0 and ω_(b)=0 rad/s, no AC), with ω_(x)=1.5302 Hz andω_(y)=0.3756 Hz; FIG. 15B—DC current for 2×10⁴ s, with ω_(x)=1.5302 Hzand ω_(y)=0.3756 Hz, and AC current with ω_(b)=0.3 rad/s, and ϕ=0 rad,and β=0.23; FIG. 15C—DC current for 2×10⁴ s, with ω_(x)=1.5302 Hz andω_(y)=0.3756 Hz, and AC current with ω_(b)=0.5 rad/s, and ϕ=0 rad, andβ=0.23.

FIG. 16 depicts a graph of the oscillations in an exemplary aluminum potmodel for a different combination of β and ω_(b) at a stabilityparameter a=1.1354 (DC current only: ω_(b)=0 rad/s and β=0, andω_(x)=1.5302 Hz and ω_(y)=0.3756 Hz.

FIG. 17 depicts simulated deviation from the average interface.

FIGS. 18A-18H depict a simulated resonant instability. FIGS. 18A-18Ddepict an interface deformation that evolves as a circulating wave. FIG.18E depicts a root-mean-square displacement from the average interfaceshape, as it varies over time. The times plotted in a-d are marked withdots. An exponential fit is plotted as a solid line. FIG. 18F depictsthe spectral power C_(nm) for all modes, with the (n;m)=(2; 0) and(0; 1) modes the strongest. FIG. 18G depicts a time evolution of thestrengths of the (n;m)=(2; 0), (0; 1), and (0; 2) modes. FIG. 18Hdepicts the spectral power of the RMS displacement, compared tofrequencies of a few gravity-wave modes, known from theory.

FIGS. 19A-19H depict a simulation showing preventing the resonantinstability by adding an oscillatory component to the electricalcurrent. FIGS. 19A-19D depict an interface deformation that evolves as astanding wave. FIG. 19E depicts a root-mean-square displacement from theaverage interface shape, as it varies over time. The times plotted ina-d are marked with dots. FIG. 19F depicts the spectral power C_(nm) forall modes with the (n;m)=(2; 0) and (0; 1) modes the strongest. FIG. 19Gdepicts a time evolution of the strengths of the (n;m)=(2; 0), (0; 1),and (0; 2) modes. FIG. 19H depicts the spectral power of the RMSdisplacement, compared to frequencies of a few gravity-wave modes, knownfrom theory.

DETAILED DESCRIPTION

The present invention can be understood more readily by reference to thefollowing detailed description, examples, drawings, and claims, andtheir previous and following description. However, before the presentarticles, systems, and/or methods are disclosed and described, it is tobe understood that this invention is not limited to the specific orexemplary aspects of articles, systems, and/or methods disclosed unlessotherwise specified, as such can, of course, vary. It is also to beunderstood that the terminology used herein is for the purpose ofdescribing particular aspects only and is not intended to be limiting.

The following description of the invention is provided as an enablingteaching of the invention in its best, currently known aspect. To thisend, those skilled in the relevant art will recognize and appreciatethat many changes can be made to the various aspects of the inventiondescribed herein while still obtaining the beneficial results of thepresent invention. It will also be apparent that some of the desiredbenefits of the present invention can be obtained by selecting some ofthe features of the present invention without utilizing other features.Accordingly, those of ordinary skill in the pertinent art will recognizethat many modifications and adaptations to the present invention arepossible and may even be desirable in certain circumstances and are apart of the present invention. Thus, the following description is againprovided as illustrative of the principles of the present invention andnot in limitation thereof.

Definitions

As used herein, the singular forms “a,” “an,” and “the” include pluralreferents unless the context clearly dictates otherwise. Thus, forexample, reference to a “device” includes aspects having two or moresuch devices unless the context clearly indicates otherwise.

Ranges can be expressed herein as from “about” one particular valueand/or to “about” another particular value. When such a range isexpressed, another aspect includes from the one particular value and/orto the other particular value. Similarly, when values are expressed asapproximations, by use of the antecedent “about,” it will be understoodthat the particular value forms another aspect. It should be furtherunderstood that the endpoints of each of the ranges are significant bothin relation to the other endpoint, and independently of the otherendpoint.

As used herein, the terms “optional” or “optionally” mean that thesubsequently described event or circumstance may or may not occur, andthat the description includes instances where said event or circumstanceoccurs and instances where it does not.

It is also to be understood that the terminology used herein is for thepurpose of describing particular aspects only and is not intended to belimiting. As used in the specification and in the claims, the term“comprising” can include the aspects “consisting of” and “consistingessentially of.”

For the terms “for example” and “such as,” and grammatical equivalencesthereof, the phrase “and without limitation” is understood to followunless explicitly stated otherwise.

As used herein, the term “substantially,” in, for example, the context“substantially no change” refers to a phenomenon or an event thatexhibits less than about 1% change, e.g., less than about 0.5%, lessthan about 0.1%, less than about 0.05%, or less than about 0.01% change.For example, when the term substantially no change is used in thecontext of substantially no change is observed in the oscillations ofthe molten electrolyte, it is understood that the change in theoscillations is less than about 1%, less than about 0.5%, less thanabout 0.1%, less than about 0.05%, or less than about 0.01%.

As used herein, the term “substantially,” in, for example, the context“substantially identical” or “substantially similar” refers to a methodor a system, or a component that is at least about 80%, at least about85%, at least about 90%, at least about 91%, at least about 92%, atleast about 93%, at least about 94%, at least about 95%, at least about96%, at least about 97%, at least about 98%, at least about 99%, orabout 100% by similar to the method, system, or the component it iscompared to.

As used herein, the terms “substantially identical reference system” or“substantially identical reference method” refer to a reference systemor method comprising substantially identical components or method stepsin the absence of an inventive component or a method step. In anotherexemplary embodiment, the term “substantially,” in, for example, thecontext “substantially identical reference systems,” refers to areference system or a method step that comprises substantially identicalcomponents or method steps, and wherein an inventive component or amethod step is substituted with a common in the art component or amethod step. For example, the term substantially identical referencesystem without the presence of AC refers to the system that has the samecomponents with the exception of the device configured to provide analternating current. In yet another example, the term substantiallyidentical reference system without the presence of AC refers to thesystem that has the same components with the alternating current presentin the system.

As used herein, the term or phrase “parametric instability” refers to aviolent undesired motion that results from the variation of a parameterwhose values the system's energy depends on.

As used herein, the term “traveling wave” can be defined mathematicallyas a surface motion in which the deformation of the surface depends onposition x and time t only in the combination k·x±ωt, where k is thewave vector, and ω is the angular frequency. In still further aspects,the term “standing wave” can be defined as a surface motion produced bythe superposition of two traveling waves, one with dependence k·x+ωt,and one with dependence k·x−ωt.

The term “traveling wave” can also be referred to as a surface motion inwhich peaks and troughs (local maxima and minima of surface height) moveas if they were traveling smoothly from place to place; while the term“standing wave” can be defined as a surface motion in which peaks andtroughs change over time by growing and shrinking in place, withouttraveling from place to place.

Numerous other general purpose or special purpose computing devicesenvironments or configurations can be used. Examples of well-knowncomputing devices, environments, and/or configurations that can besuitable for use include, but are not limited to, personal computers,server computers, handheld or laptop devices, smartphones,multiprocessor systems, microprocessor-based systems, network personalcomputers (PCs), minicomputers, mainframe computers, embedded systems,distributed computing environments that include any of the above systemsor devices, and the like.

Computing processors or devices, as disclosed herein, can containcommunication connection(s) that allow the device to communicate withother devices. Computing devices can also have input device(s) such as akeyboard, mouse, pen, voice input device, touch input device, etc.Output device(s) such as a display, speakers, printer, etc., can also beincluded. All these devices are well known in the art and need not bediscussed at length here.

Computer-executable instructions, such as program modules being executedby a computer, can be used. Generally, program modules include routines,programs, objects, components, data structures, etc., that performparticular tasks or implement particular abstract data types.Distributed computing environments can be used where tasks are performedby remote processing devices that are linked through a communicationsnetwork or other data transmission medium. In a distributed computingenvironment, program modules and other data can be located in both localand remote computer storage media, including memory storage devices.

In its most basic configuration, a computing device typically includesat least one processing unit and memory. Depending on the exactconfiguration and type of computing device, memory can be volatile (suchas random-access memory (RAM)), non-volatile (such as read-only memory(ROM), flash memory, etc.), or some combination of the two.

Computing devices can have additional features/functionality. Forexample, a computing device can include additional storage (removableand/or non-removable) including, but not limited to, magnetic or opticaldisks or tape.

Computing device typically includes a variety of computer-readablemedia. Computer-readable media can be any available media that can beaccessed by the device and includes both volatile and non-volatilemedia, removable and non-removable media.

Computer storage media include volatile and non-volatile, and removableand non-removable media implemented in any method or technology forstorage of information such as computer-readable instructions, datastructures, program modules or other data. Memory, removable storage,and non-removable storage are all examples of computer storage media.Computer storage media include, but are not limited to, RAM, ROM,electrically erasable program read-only memory (EEPROM), flash memory orother memory technology, CD-ROM, digital versatile disks (DVD) or otheroptical storage, magnetic cassettes, magnetic tape, magnetic diskstorage or other magnetic storage devices, or any other medium which canbe used to store the desired information and which can be accessed by acomputing device. Any such computer storage media can be part of acomputing device.

As disclosed herein, computing devices can contain communicationconnection(s) that allow the device to communicate with other devices.The connection can be wireless or wired. Computing devices can also haveinput device(s) such as a keyboard, mouse, pen, voice input device,touch input device, etc. Output device(s) such as a display, speakers,printer, etc., can also be included. All these devices are well known inthe art and need not be discussed at length here.

It should be understood that the various techniques described herein canbe implemented in connection with hardware components or softwarecomponents or, where appropriate, with a combination of both.Illustrative types of hardware components that can be used includeField-programmable Gate Arrays (FPGAs), Application-specific IntegratedCircuits (ASICs), Application-specific Standard Products (ASSPs),System-on-a-chip systems (SOCs), Complex Programmable Logic Devices(CPLDs), etc. The methods and apparatus of the presently disclosedsubject matter, or certain aspects or portions thereof, can take theform of program code (i.e., instructions) embodied in tangible media,such as CD-ROMs, hard drives, or any other machine-readable storagemedium where, when the program code is loaded into and executed by amachine, such as a computer, the machine becomes an apparatus forpracticing the presently disclosed subject matter. In yet other aspects,the software can comprise any simulation software that can be used orapplied for electrolytic cells. For example, it can comprise MHD-VALDISsimulation software. However, this simulation software is only optional,exemplary and non-limiting.

While aspects of the present invention can be described and claimed in aparticular statutory class, such as the system statutory class, this isfor convenience only and one of ordinary skill in the art willunderstand that each aspect of the present invention can be describedand claimed in any statutory class. Unless otherwise expressly stated,it is in no way intended that any method or aspect set forth herein beconstrued as requiring that its steps be performed in a specific order.Accordingly, where a method claim does not specifically state in theclaims or descriptions that the steps are to be limited to a specificorder, it is in no way intended that an order be inferred in anyrespect. This holds for any possible non-express basis forinterpretation, including matters of logic with respect to arrangementof steps or operational flow, plain meaning derived from grammaticalorganization or punctuation, or the number or type of aspects describedin the specification.

The present invention may be understood more readily by reference to thefollowing detailed description of various aspects of the invention andthe examples included therein and to the Figures and their previous andfollowing description.

Methods

The present disclosure is directed to a method comprising applying analternating current (AC) comprising an oscillatory current waveform toan electrolytic cell comprising an electrolyte for a first predeterminedtime, wherein waveform comprises an amplitude, frequency and/or phasethat are predetermined to stabilize the electrolytic cell such thatsubstantially no change in a current oscillation is observed in theelectrolyte during electrolysis.

It is understood that the method disclosed herein can be applied to anyelectrolytic cell known in the art. Many various metals are currentlyproduced by utilizing high capacity electrolytic cells. For example, andwithout limitations, the methods of the current disclosure can beapplied to an aluminum electrolytic cell, iron electrolytic cell steelelectrolytic cell, titanium electrolytic cell, and the like. In yetfurther aspects, the methods disclosed herein are applied to thealuminum electrolytic cell.

It is further understood that the specific electrolyte can bespecifically chosen for the desired electrolytic cell. In some aspects,the electrolyte can be an ionic solution. Yet, in other aspects, theelectrolyte can comprise ionic liquids. While in still other aspects,the electrolyte can comprise molten salts.

In yet other aspects, any molten salt electrolytes can be used. In someexemplary aspects, the molten salt electrolyte comprises cryolite. Incertain aspects, cryolite is a naturally occurring mineral comprisingNa₂NaAlF₆. In yet other aspects, the cryolite is synthetically produced.Cryolite is known to dissolve alumina oxide. In yet further aspects, themolten salt electrolyte can further comprise aluminum fluoride, alumina,or a combination thereof. In yet further aspects, lithium fluoride andmagnesium fluoride can be added in the form of lithium carbonate andmagnesium oxide to form the molten salt electrolyte. It is understoodthat in aspects where lithium fluoride is present, lithium fluoride canparticipate in lowering the melting point of the salt electrolyte,decreasing the vapor pressure, density, reduced species solubility andviscosity, and it can also increase electrical conductivity.

Currently, aluminum production through electrolysis consumes about 3% ofelectricity worldwide (P. A. DAVIDSON and R. I. LINDSAY. Stability ofinterfacial waves in aluminum reduction cells. Journal of FluidMechanics, 362:273-295, 1998). In general, aluminum is produced byrunning large, steady (DC) electrical current through pools of moltensalts atop liquid aluminum. About 40% of that energy is wasted as heatin the salt and could be saved if the salt layer were thinner.Unfortunately, thinning the salt results in a resonant instability onthe interface of the salt and liquid aluminum, in which surface wavesgrow uncontrollably until the smelter is shutdown.

The resonant instability deforms the aluminum-salt interface in a waythat changes overtime, and when the deformation becomes too large, thecell must be shut down.

It is understood that even without the presence of instability, theinterface between the molten salt and liquid aluminum is not flat.Rather, it has a dome shape, caused by the combined effects of thesteady electrical current and bubbles produced in aluminum'selectrochemical production. Thus, such commonly used methods are energyconsuming and inefficient.

In some aspects, in the methods disclosed herein, a direct current (DC)can be applied to electrolytic cell for a second predetermined timeprior to applying the AC. While in other aspects, the direct current iscontinued to be applied for the first predetermined time simultaneouslywith applying the AC. It is understood that the first predetermined timecan be any time needed to accomplish the desired electrolytic process.This time can be determined by the desired amount of metal to beproduced by the electrolytic process, or by the amount of electrolytepresent in the cell, or by the maintenance requirements of theelectrolytic cell

In some aspects, the predetermined period of time can be any time periodrequired to achieve a desirable result. In certain aspects, thepredetermined period of time can be from about 1 sec to about 72 h,including exemplary values of about 20 sec, about 60 sec, about 5 min,about 10 min, about 30 min, about 1 h, about 5 h, about 10 h, about 15h, about 20 h, about 24 h, about 30 h, about 36 h, about 42 h, about 48h, about 54 h, about 60 h, about 66 h, and about 70 h. In yet furtheraspects, the electrolytic cell can exhibit substantially no change inthe oscillations present in the molten salt during the electrolysis whenthe AC is provided.

In still further aspects, when the AC is not provided to the cellsimultaneously with the DC, the DC can be applied to the cell for thesecond predetermined time, before the AC is applied. This secondpredetermined time can be defined by a time period measured fromapplying the DC to an appearance of a resonant instability in theelectrolytic cell. It is understood that in such aspects, when theresonant instability appears in the electrolytic cell, the addition ofthe AC according to the methods disclosed herein stabilizes the cell,and the cell can operate as long as desired.

The electrolytic cell used in the methods disclosed herein can furthercomprise an anode and a cathode. It is understood that any known in theart anodes and cathodes can be utilized depending on the specificelectrolytic cell. In certain aspects, the anode and/or cathode cancomprise carbon.

In the methods disclosed herein, the AC properties such as theamplitude, frequency and/or phase can be predetermined such that ananode-to-cathode distance is reduced when it is compared to ananode-to-cathode distance (ACD) of a substantially identical referenceelectrolytic cell in the absence of applying an AC. As shown in theExamples below, the conventional electrolytic cell, in the absence ofthe AC, can operate without sufficient instability when the ACD is notsmaller than 4.2 cm. Any attempts to decrease this distance in theconventional cells can result in the appearance of instability andrendering the electrolytic cell un-operational.

The methods disclosed herein allow reduction of the ACD value whilekeeping the electrolytic cell stable with substantially no change in thecurrent oscillation during the electrolysis.

It is understood that in some aspects, the methods disclosed hereinallow to obtain an electrolytic cell that exhibits an increase in theenergy efficiency when is compared to the energy efficiency of asubstantially identical reference electrolytic cell in the absence ofapplying the AC. In certain aspects, the electrolytic cell exhibits anincrease in energy efficiency that is substantially proportional to areduction in the anode-to-cathode distance. In some exemplary andunlimiting aspects, if the ACD is reduced by about 5%, the energyefficiency can be increased by about 5%.

In still further aspects, the methods disclosed herein allow theelectrolytic cell to exhibit an increase in the energy efficiency of atleast about 5%, at least about 7%, at least about 10%, at least about12%, or at least about 14% when compared to the substantially identicalreference electrolytic cell in the absence of applying an AC.

In still further aspects, the AC has a waveform that is defined by aplurality of modes. In such aspects, the combination of the plurality ofmodes can result in the formation of a standing wave. In still furtheraspects, the plurality of modes of the waveform are configured todisrupt the formation and/or growth of circulating waves that areresponsible for the electrolytic cell instability.

In certain aspects, the oscillatory current waveform can comprise anyshape. In some exemplary and unlimiting aspects, the oscillatory currentwaveform can comprise a sinusoidal, cosinusoidal, triangular, or squareshape.

In certain aspects, the AC can be provided by any known in the artmethods. In some aspects, the AC can be provided by a device comprisingan alternating current source that is in electrical communication withthe electrolytic cell. Any known in the art AC devices can be utilized.In some exemplary and unlimiting aspects, the device can comprisesolid-state components, mechanical components, or any other componentscapable of providing an alternating current. As disclosed in detailabove, the device can generate an AC oscillatory current waveformdefined by the desired predetermined amplitude, frequency, and/or phase.

In yet other aspects, and as described above, the DC can also beprovided by any known in the art sources. In some aspects, the DC isprovided by the same device that is providing the AC. In such aspects,in addition to the alternating current source, the device can comprise adirect current source. In yet other aspects, the DC can be provided by aseparate direct current source. It is understood that any of the DCsources can provide DC having the desired amplitude and in an amountneeded to imitate the electrolytic reaction.

In still further aspects, it is understood that the predeterminedamplitude of the DC can be chosen based on the specific parameters ofthe electrolytic cell and the desired results. In certain aspects, thepredetermined amplitude of the DC has an amplitude effective to initiateelectrolysis. In yet further aspects, the predetermined amplitude of thedirect current can have values from about 100 kA to about 500 kA,including exemplary values of about 120 kA, about 150 kA, about 180 kA,about 200 kA, about 220 kA, about 250 kA, about 280 kA, about 300 kA,about 320 kA, about 350 kA, about 380 kA, about 400 kA, about 420 kA,about 450 kA, and about 480 kA. It is further understood that any valuebetween any two foregoing values can be used for the direct current.

In still further aspects, the predetermined amplitude, frequency, and/orphase of the AC can be chosen based on the specific parameters of thecell. In yet other aspects, it can be selected such that theelectrolytic cell having the AC applied to exhibit no substantialoscillations in the molten electrolyte during the electrolysis.

In certain aspects, the predetermined amplitude of the AC can havevalues from about 0.5 kA to about 50 kA, including exemplary values ofabout 1 kA, about 1.5 kA, about 2 kA, about 2.5 kA, about 3 kA, about3.5 kA, about 4 kA, about 4.5 kA, about 5 kA, about 5.5 kA, about 6 kA,about 6.5 kA, about 7 kA, about 7.5 kA, about 8 kA, about 8.5 kA, about9 kA, about 9.5 kA, about 10 kA, about 12 kA, about 15 kA, about 17 kA,about 20 kA, about 22 kA, about 25 kA, about 27 kA, about 30 kA, about32 kA, about 35 kA, about 37 kA, about 40 kA, and about 42 kA, about 45kA, and about 47 kA. It is further understood that any value between anytwo foregoing values can be used for the AC

In yet further aspects, the AC can be further defined by anon-dimensional amplitude ratio β of the predetermined amplitude of theAC to the predetermined amplitude of the DC, wherein the β is fromgreater than 0 to about 1, including an exemplary value of about 0.1,about 0.2, about 0.3, about 0.4, about 0.5, about 0.6, about 0.7, about0.8, and about 0.9. Yet, in other aspects, the β is from greater than 0to about 0.15, including exemplary value of about 0.05, about 0.07,about 0.1, about 0.11, about 0.12, about 0.13, and about 0.14. It isunderstood that the non-dimensional amplitude ratio β can have any valuebetween any two foregoing values.

In still further aspects, the predetermined frequency of the AC can befrom greater than 0 to about π rad/s, including exemplary values ofabout π/50 rad/s, about π/40 rad/s, about π/30 rad/s, about π/20 rad/s,about π/10 rad/s, and π/5 rad/s. It is understood that the predeterminedfrequency of the AC can have any value between any two foregoing values.

In still further aspects, the predetermined frequency of the AC can befrom greater than 0 Hz to about 0.5 Hz, including exemplary values ofabout 0.01 Hz, about 0.02 Hz, about 0.03 Hz, about 0.04 Hz, about 0.05Hz, about 0.06 Hz, about 0.07 Hz, about 0.08 Hz, about 0.09 Hz, about0.1 Hz, about 0.15 Hz, about 0.2 Hz, about 0.25 Hz, about 0.3 Hz, about0.35 Hz, about 0.4 Hz, and about 0.45 Hz. It is understood that thepredetermined frequency of the AC can have any value between any twoforegoing values.

In still further aspects, the predetermined frequency of the AC can alsobe from greater than 0 to about 120π rad/s, including exemplary valuesof about 1/3π rad/s, about 1/2π rad/s, about 3/4π rad/s, about π rad/s,about 3π/2 rad/s, about 2π rad/s, about 3π rad/s, about 4π rad/s, about5π rad/s, about 10π rad/s, about 15π rad/s, about 20π rad/s, about 25πrad/s, about 30π rad/s, about 35π rad/s, about 40π rad/s, about 45πrad/s, about 50π rad/s, about 55π rad/s, about 60π rad/s, about 65πrad/s, about 70π rad/s, about 75π rad/s, about 80π rad/s, about 85πrad/s, about 90π rad/s, about 95π rad/s, about 100 π rad/s, about 105πrad/s, about 110π rad/s, and about 115π rad/s. It is understood that thepredetermined frequency of the AC can have any value between any twoforegoing values.

In still further aspects, the predetermined frequency of the AC can befrom greater than 0 Hz to about 60 Hz, including exemplary values ofabout 0.1 Hz, about 0.5 Hz, about 1 Hz, about 2 Hz, about 5 Hz, about 10Hz, about 20 Hz, about 30 Hz, about 40 Hz, and about 50 Hz. It isunderstood that the predetermined frequency of the AC can have any valuebetween any two foregoing values.

In yet further aspects, the AC can have the predetermined β values fromgreater than 0 to about 0.15, including exemplary values of about 0.05,about 0.07, about 0.1, about 0.11, about 0.12, about 0.13, and about0.14; and the predetermined frequency values from greater than 0 toabout π rad/s, including exemplary values of about π/50 rad/s, aboutπ/40 rad/s, about π/30 rad/s, about π/20 rad/s, about π/10 rad/s, andπ/5 rad/s. In still further aspects, the AC can have any β values andany frequency values between any two foregoing values.

In yet further aspects, the AC can have the predetermined β values fromgreater than 0 to about 0.15, including exemplary values of about 0.05,about 0.07, about 0.1, about 0.11, about 0.12, about 0.13, and about0.14; and the predetermined frequency values from greater than 0 toabout 0.5 Hz, including exemplary values of about 0.01 Hz, about 0.02Hz, about 0.03 Hz, about 0.04 Hz, about 0.05 Hz, about 0.06 Hz, about0.07 Hz, about 0.08 Hz, about 0.09 Hz, about 0.1 Hz, about 0.15 Hz,about 0.2 Hz, about 0.25 Hz, about 0.3 Hz, about 0.35 Hz, about 0.4 Hz,and about 0.45 Hz. In still further aspects, the AC can have any βvalues and any frequency values between any two foregoing values.

In yet further aspects, the AC can have the predetermined β values fromgreater than 0 to about 1, about 0.1, including exemplary values ofabout 0.2, about 0.3, about 0.4, about 0.5, about 0.6, about 0.7, about0.8, and about 0.9; and the predetermined frequency values from greaterthan 0 to about 120n rad/s, including exemplary values of about 1/3πrad/s, about 1/2π rad/s, about 3/4π rad/s, about π rad/s, about 3π/2rad/s, about 2π rad/s, about 3π rad/s, about 4π rad/s, about 5π rad/s,about 10π rad/s, about 15π rad/s, about 20π rad/s, about 25π rad/s,about 30π rad/s, about 35π rad/s, about 40π rad/s, about 45π rad/s,about 50π rad/s, about 55π rad/s, about 60π rad/s, about 65π rad/s,about 70π rad/s, about 75π rad/s, about 80π rad/s, about 85π rad/s,about 90π rad/s, about 95π rad/s, about 100 π rad/s, about 105π rad/s,about 110π rad/s, and about 115π rad/s. In still further aspects, the ACcan have any β values and any frequency values between any two foregoingvalues.

In yet further aspects, the AC can have the predetermined β values fromgreater than 0 to about 1, about 0.1, including exemplary values ofabout 0.2, about 0.3, about 0.4, about 0.5, about 0.6, about 0.7, about0.8, and about 0.9; and the predetermined frequency values from greaterthan 0 to about 60 Hz, including exemplary values of about 0.1 Hz, about0.5 Hz, about 1 Hz, about 2 Hz, about 5 Hz, about 10 Hz, about 20 Hz,about 30 Hz, about 40 Hz, and about 50 Hz. In still further aspects, theAC can have any β values and any frequency values between any twoforegoing values.

It is further understood that the standing waves can be affected by theviscosity and/or surface tension of the electrolyte, and therefore theseparameters need to be taken into consideration when the AC parametersare chosen for the specific cell.

In still further aspects, the methods disclosed herein allowsubstantially no change in oscillations to be present in the molten saltelectrolyte over a predetermined period of time when the AC is providedto the electrolytic cell. In such aspects, the methods disclosed hereinallow to reduce the anode-to-cathode distance relative to theconventional methods. In some aspects, the anode-to-cathode distance isalso equivalent to the thickness of the electrolyte layer. In suchexemplary aspects, the anode-to-cathode distance or the thickness of theelectrolyte can be equal to or less than about 4.5 cm, less than about4.3 cm, less than about 4.2 cm, less than about 4 cm, less than about3.8 cm, less than about 3.5 cm, less than about 3.3 cm, less than about3 cm, less than about 2.8 cm, less than about 2.5 cm, or less than about2 cm. In still further aspects, the thickness of the electrolyte or theanode-to-cathode distance can be anywhere between about 3 cm to about4.3 cm, including exemplary values of about 3.1 cm, about 3.2 cm, about3.3 cm, about 3.4 cm, about 3.5 cm, about 3.6 cm, about 3.7 cm, about3.8 cm, about 3.9 cm, about 4.0 cm, about 4.1 cm, and about 4.2 cm.

In still further aspects, the methods disclosed herein comprise a stepof measuring the current oscillations. It is understood that the step ofmeasuring can be performed by any known in the art methods. In someexemplary aspects, the measuring can be performed with by a controllingunit that is in a feedback loop communication with the device and theelectrolytic cell such that it is configured to receive an inputcommunication comprising a first data from the device and/or theelectrolytic cell and provide an output communication to the deviceand/or the electrolytic cell, wherein the output communication comprisesa second data adjusted for the first data.

In such an aspect, the controlling unit collects and records the firstdata, wherein the first data can comprise physical and electricalparameters of the cell. The signal obtained from the cell and/or deviceis evaluated. The controlling unit then can communicate to the deviceand/or the electrolytic cell the second data that is adjusted for thespecific cell/device parameters to obtain the desired properties. Insuch aspects, the second data can comprise the predetermined amplitude,frequency, and/or phase, or β of AC based on the feedback communicationfrom the electrolytic cell and/or device. In still further aspects, thedevice is also in a feedback loop communication with the electrolyticcell.

In yet further aspects, the methods disclosed herein comprise a step ofadjusting the predetermined amplitude, frequency, and/or phase, or β ofAC based on the feedback communication from the electrolytic cell.

It is understood that the data collection, its analysis and adjustmentcan be performed by a computing processor or a computing device. Forexample, the controlling unit can comprise a computing processor. Yet,in other examples, both or either the electrolytic cell and device cancomprise a computing processor. In still further aspects, thecontrolling unit, device, and electrolytic cell can be in communicationwith the same computing processor. Any known in the art computingprocessors capable of the desired task can be used. In still furtheraspects, the data communicated to the computing processor can beprovided by any means, including wireless communication, wiredcommunication, through intermediate media such as a flash drive, aCD-ROM or a DVD.

In still further aspects, the controlling unit can also comprise ameasuring unit that is configured to measure the specific output of theelectrolytic cell or that of the device. In such aspects, the measuringunit can be configured, for example, to measure an electrical responsefrom the cell and can comprise ultrasound probes, laser range finders,capacitive probes, and the like.

Also disclosed herein is a method comprising: a) providing a first datato a computational processor, wherein the first data comprises at leastone of one or more of geometric parameters of an electrolytic cell, acathode-to-anode-distance of the electrolytic cell, a value of a directcurrent; an amplitude of a direct current, a thickness of a metal layer,material properties of a metal, material properties of an electrolyte,material properties of a cathode, material properties of an anode, orany combination thereof; b) analyzing the first data by thecomputational processor to provide a second data comprising parametersof an alternating current (AC) wherein the parameters comprise one ormore of a first amplitude, a first frequency, and/or a first phase of anoscillatory current form of the AC; and c) applying the AC having one ormore parameters present in the second data to the electrolytic cell tostabilize the electrolytic cell.

In still further aspects, the methods can further comprise d) collectinga third data from the electrolytic cell and transferring the third datato the computational processor to analyze the performance of theelectrolytic cell; e) analyzing the third data by the computationalprocessor to provide a fourth data comprising parameters of thealternating current (AC) wherein the parameters comprise one or more ofa second amplitude, a second frequency, and/or a second phase of anoscillatory current form of the AC; and f) applying the AC having one ormore parameters present in the fourth data to the electrolytic cell.

It is understood that in such aspects, the first data can comprise anydata that relates to a specific electrolytic cell. The methods disclosedherein can be used to provide tuned AC parameters as a function of theparameters of the specific electrolytic cell. It is also understood thatany of the mentioned above parameters can be used. In some aspects, allof the above parameters can be provided to the computational processorfor analysis and determination of the specific AC parameters.

In still further aspects, after the computational processor provides thesecond data with the desired AC parameters, the AC is applied to theelectrolytic cell. The electrolytic cell is then monitored, for example,for stability and/or energy performance, the third data from theelectrolytic cell can be transferred back to the computational processorfor analysis of the cell performance. If needed, the AC parameters canbe adjusted if needed, and a new AC having adjusted parameters can beapplied to the electrolytic cell. It is further understood that thesteps of receiving the data from the electrolytic cell, analyzing it andproviding adjusted parameters for AC to be applied to the electrolyticcell can be repeated as many times as needed to ensure that theelectrolytic cell is stabilized such that substantially no change in acurrent oscillation is observed in an electrolyte during electrolysis.

It is also understood that in the aspects where DC is provided prior tothe applying AC, the computational processor can also include this datain addition to any other data mentioned above and provide the operatorwith timing when the AC should be applied to the cell.

Also disclosed herein are the methods for increasing energy efficiencyin an electrolytic cell. In such aspects, the methods can compriseapplying an alternating current (AC) comprising an oscillatory currentwaveform to the electrolytic cell comprising an electrolyte for a firstpredetermined time, wherein waveform comprises an amplitude, frequencyand/or phase that are predetermined to stabilize the electrolytic cellsuch that substantially no change in a current oscillation is observedin the electrolyte during electrolysis; wherein the energy efficiency isincreased by at least about 5% when compared to a substantiallyidentical reference electrolytic cell in the absence of applying an AC.In such aspects, the energy efficiency can also be increased by at leastabout 7%, at least about 8%, at least about 9%, at least about 10%, atleast about 11%, at least about 12%, at least about 13%, at least about14%, at least about 15%, at least about 16%, at least about 17%, atleast about 18%, at least about 19%, or even at least about 20% whencompared to a substantially identical reference electrolytic cell in theabsence of applying an AC.

Systems

In some aspects described herein is a system comprising: a) anelectrolytic cell comprising: i) an anode; ii) a cathode; and iii) amolten electrolyte having a predetermined thickness; b) a direct currentsource that is in electrical communication with the electrolytic celland is configured to provide a direct current (DC) having apredetermined amplitude and to initiate an electrolysis reaction in theelectrolytic cell; c) a device comprising an alternating current source(AC); wherein the device is in electrical communication with theelectrolytic cell and is configured to provide an alternating current(AC) to the electrolytic cell, wherein the AC comprises an oscillatorycurrent waveform defined by a predetermined amplitude, frequency, and/orphase; and wherein the device is in feedback loop communication with theelectrolytic cell; and wherein the electrolytic cell exhibitssubstantially no change in oscillations present in the molten saltelectrolyte over a predetermined period of time when the AC is providedto the electrolytic cell.

It is understood that any of the electrolytic cells mentioned above canbe utilized. In still further exemplary aspects, the disclosedelectrolytic cell is an aluminum electrolysis cell. In such cells, themolten salt electrolyte can comprise cryolite.

In still further exemplary aspects, the anode comprises a carbon. In yetother aspects, the cathode comprises a carbon. In still further aspects,both anode and cathode comprise carbon blocks.

An exemplary and unlimiting aluminum production cell is shown in FIG. 1. Aluminum is produced using Hall-Heroult cells (H-H cells) 100, whichhas two large carbon blocks that serve as electrodes (an anode (102) onthe top of the cell and cathode (104) at the bottom) with two liquidlayer in between (aluminum (108) at the bottom, electrolyte on top(106)) (UC RUSAL. How aluminium is produced, 2019). H-H cells utilize anelectrochemical process known as electrolysis. Electrolysis useselectrical currents to separate elements from naturally occurring ores.In the Hall-Heroult process, naturally occurring alumina (aluminumoxide) is first dissolved in a bath of molten electrolyte (cryolite).Applying a direct electrical current (DC), typically greater than 300kA, decomposes the naturally occurring alumina into aluminum metal(deposited at the cathode) and carbon dioxide gas (produced at theanode) (UC RUSAL. How aluminium is produced, 2019)

However, only about 60% is consumed for that purpose as the remaining40% becomes heat through a process known as Joule heating (resistiveheating) (P. A. Davidson. Overcoming instabilities in aluminiumreduction cells: a route to cheaper aluminium. Materials Science andTechnology, 2000). Joule heating is caused by electrons from thesupplied current interacting with the atoms of the conducting materialand scales as the supplied current (J) squared multiplied by theelectrical resistance (R) of the conducting material. The resistance Rdepends on a material property known as electrical resistivity. For theH-H cell, the molten cryolite layer has an electrical resistivity 100times bigger than that of the carbon blocks and 10,000 times bigger thanthat of the molten aluminum layer, making it the dominant source ofelectrical energy lost to heat. Without wishing to be bound by anytheory, it is hypothesized that a decrease in the height of theelectrolyte layer can result in reduced energy loss. However, currently,when the height of the molten electrolyte layer is reduced below athreshold of around 4.5 cm, the molten layers inside the cell sloshviolently, sloshing back and forth, which can result in the aluminumlayer touching the carbon anode at the top, and therefore, in failure toelectrolyze alumina.

This sloshing is an example of parametric instability. In the case ofthe sloshing instability, small gravity-restored waves at the interface(110) between the aluminum and cryolite layers, which occur naturally,are amplified strongly by the electrical current in a feed-forwardprocess. The difference in resistivity between the two layers draws morecurrent into regions where the cryolite is thin and less into regionswhere it is thick. Without wishing to be bound by any theory, it isassumed that this will result in a compensating horizontal current inthe aluminum layer that interacts with vertical magnetic fields fromnearby cells to produce forces that make the thin regions thinner andthe thick regions thicker (P. A. DAVIDSON and R. I. LINDSAY. Stabilityof interfacial waves in aluminium reduction cells. Journal of FluidMechanics, 362:273-295, 1998). More current is drawn into thin regions,and the cycle repeats. It is understood that existing technology doesnot allow a reduction of the molten electrolyte layer below ˜4.5 cm, assuch a reduction renders an H-H cell unstable.

In certain aspects, and as disclosed herein, to improve the stability ofthe H-H cell and to minimize parametric instability, the disclosedsystem comprises an oscillatory (AC) component added to the electricalcurrent.

In further aspects, and as disclosed herein, the system comprises adevice comprising an alternating current source; wherein the device isin electrical communication with the electrolytic cell and is configuredto provide an alternating current (AC) to the electrolytic cell, whereinthe AC comprises an oscillatory current waveform defined by apredetermined amplitude, frequency, and/or phase. In yet other aspects,the direct current source that is in electrical communication with theelectrolytic cell and configured to provide a direct current (DC) havinga predetermined amplitude and to initiate an electrolysis reaction inthe electrolytic cell is present in the same device, as the AC source.However, in other aspects, the DC source can be present in a separatedevice.

In still further aspects, the system can further comprise a controllingunit configured to measure oscillations of the molten salt electrolyteas a function of the DC and AC applied to the electrolytic cell, andwherein the controlling unit is in a feedback loop communication withthe device and the electrolytic cell.

FIG. 2 shows an exemplary system 200 that comprises an aluminum pot(202) that is in electrical communication with the device 204. In thisexemplary aspect, the device 204 comprises both AC and DC sources. TheAC and DC are provided to the aluminum pot 202 by the electricalconnection 201. The electrical response of the cell is measured (203) bya measuring unit (206). The response can then be fed back (205) to thedevice 204 to adjust the current parameters based on the desiredoutcome.

FIG. 3 shows an exemplary system 300 in another aspect. In such anaspect, the system can comprise an aluminum pot (302) that is inelectrical communication with the device 308 that is configured toprovide a direct current and the device 304 that is configured toprovide an alternating current. The electrical response of the cell 302is measured (303) by a measuring unit (306). The response can then befed back (305) to the device 304 to adjust the current parameters basedon the desired outcome.

In certain aspects, the measuring unit is configured to measure anelectrical response from the cell and can comprise ultrasound probes,laser range finders, capacitive probes, and the like. In still furtheraspects, the measuring unit can be a part of the controlling unit. Inyet other aspects, the controlling unit can also comprise a measuringunit that is in communication with the device and/or electrolytic cell.

In still further aspects, the electrolytic cell of the disclosed systemexhibits substantially no change in oscillations present in the moltensalt electrolyte over a predetermined period of time when the AC isprovided to the electrolytic cell. In yet other aspects, no change inoscillations present in the molten salt electrolyte can be observed forthe cell where the predetermined thickness of the molten electrolyte isequal to or less than about 4.5 cm, less than about 4.3 cm, less thanabout 4.2 cm, less than about 4 cm, less than about 3.8 cm, less thanabout 3.5 cm, less than about 3.3 cm, less than about 3 cm, less thanabout 2.8 cm, less than about 2.5 cm, or less than about 2 cm. In stillfurther aspects, the thickness of the electrolyte is between about 3 cmto about 4.3 cm, including exemplary values of about 3.1 cm, about 3.2cm, about 3.3 cm, about 3.4 cm, about 3.5 cm, about 3.6 cm, about 3.7cm, about 3.8 cm, about 3.9 cm, about 4.0 cm, about 4.1 cm, and about4.2 cm. In yet other aspects, the predetermined thickness of theelectrolyte is substantially identical to an anode-to-cathode-distance,as disclosed above.

In still further aspects, the disclosed systems allow a reduction in theanode-to-cathode distance. In such aspects, the amplitude, frequencyand/or phase of the AC are predetermined such that an anode-to-cathodedistance is reduced when it is compared to an anode-to-cathode distanceof a substantially identical reference electrolytic cell in the absenceof providing the AC.

In still further unlimiting aspects, the oscillatory current waveform ofAC can comprise a sinusoidal, cosinusoidal, triangular, or square shape.It is understood that the shape, normalized amplitude, normalizedfrequency, and/or phase could also change over time, for example, inresponse to feedback. In yet further aspects, the waveform is defined bya plurality of modes forming a standing wave. These standing waves areconfigured to disrupt a formation and growth of circulating waves, asdisclosed herein.

In yet further aspects, the AC can be further defined by anon-dimensional amplitude ratio β of the predetermined amplitude of theAC to the predetermined amplitude of the DC, wherein the β is fromgreater than 0 to about 1, including exemplary value of about 0.1, about0.2, about 0.3, about 0.4, about 0.5, about 0.6, about 0.7, about 0.8,and about 0.9. Yet, in other aspects, the β is from greater than 0 toabout 0.15, including exemplary value of about 0.05, about 0.07, about0.1, about 0.11, about 0.12, about 0.13, and about 0.14. It isunderstood that the non-dimensional amplitude ratio β can have any valuebetween any two foregoing values.

In still further aspects, the predetermined frequency of the AC can befrom greater than 0 to about π rad/s, including exemplary values ofabout π/50 rad/s, about π/40 rad/s, about π/30 rad/s, about π/20 rad/s,about π/10 rad/s, and π/5 rad/s. It is understood that the predeterminedfrequency of the AC can have any value between any two foregoing values.

In still further aspects, the predetermined frequency of the AC can befrom greater than 0 Hz to about 0.5 Hz, including exemplary values ofabout 0.01 Hz, about 0.02 Hz, about 0.03 Hz, about 0.04 Hz, about 0.05Hz, about 0.06 Hz, about 0.07 Hz, about 0.08 Hz, about 0.09 Hz, about0.1 Hz, about 0.15 Hz, about 0.2 Hz, about 0.25 Hz, about 0.3 Hz, about0.35 Hz, about 0.4 Hz, and about 0.45 Hz. It is understood that thepredetermined frequency of the AC can have any value between any twoforegoing values.

In still further aspects, the predetermined frequency of the AC can alsobe from greater than 0 to about 120π rad/s, including exemplary valuesof about 1/3π rad/s, about 1/2π rad/s, about 3/4π rad/s, about π rad/s,about 3π/2 rad/s, about 2π rad/s, about 3π rad/s, about 4π rad/s, about5π rad/s, about 10π rad/s, about 15π rad/s, about 20π rad/s, about 25πrad/s, about 30π rad/s, about 35π rad/s, about 40π rad/s, about 45πrad/s, about 50π rad/s, about 55π rad/s, about 60π rad/s, about 65πrad/s, about 70π rad/s, about 75π rad/s, about 80π rad/s, about 85πrad/s, about 90π rad/s, about 95π rad/s, about 100π rad/s, about 105πrad/s, about 110π rad/s, and about 115π rad/s. It is understood that thepredetermined frequency of the AC can have any value between any twoforegoing values.

In still further aspects, the predetermined frequency of the AC can befrom greater than 0 Hz to about 60 Hz, including exemplary values ofabout 0.1 Hz, about 0.5 Hz, about 1 Hz, about 2 Hz, about 5 Hz, about 10Hz, about 20 Hz, about 30 Hz, about 40 Hz, and about 50 Hz. It isunderstood that the predetermined frequency of the AC can have any valuebetween any two foregoing values.

In yet further aspects, the AC can have the predetermined β values fromgreater than 0 to about 0.15, including exemplary values of about 0.05,about 0.07, about 0.1, about 0.11, about 0.12, about 0.13, and about0.14; and the predetermined frequency values from greater than 0 toabout π rad/s, including exemplary values of about π/50 rad/s, aboutπ/40 rad/s, about π/30 rad/s, about π/20 rad/s, about π/10 rad/s, andπ/5 rad/s. In still further aspects, the AC can have any β values andany frequency values between any two foregoing values.

In yet further aspects, the AC can have the predetermined β values fromgreater than 0 to about 0.15, including exemplary values of about 0.05,about 0.07, about 0.1, about 0.11, about 0.12, about 0.13, and about0.14; and the predetermined frequency values from greater than 0 toabout 0.5 Hz, including exemplary values of about 0.01 Hz, about 0.02Hz, about 0.03 Hz, about 0.04 Hz, about 0.05 Hz, about 0.06 Hz, about0.07 Hz, about 0.08 Hz, about 0.09 Hz, about 0.1 Hz, about 0.15 Hz,about 0.2 Hz, about 0.25 Hz, about 0.3 Hz, about 0.35 Hz, about 0.4 Hz,and about 0.45 Hz. In still further aspects, the AC can have any βvalues and any frequency values between any two foregoing values.

In yet further aspects, the AC can have the predetermined β values fromgreater than 0 to about 1, about 0.1, including exemplary values ofabout 0.2, about 0.3, about 0.4, about 0.5, about 0.6, about 0.7, about0.8, and about 0.9; and the predetermined frequency values from greaterthan 0 to about 120π rad/s, including exemplary values of about 1/3πrad/s, about 1/2π rad/s, about 3/4π rad/s, about π rad/s, about 3π/2rad/s, about 2π rad/s, about 3π rad/s, about 4π rad/s, about 5π rad/s,about 10π rad/s, about 15π rad/s, about 20π rad/s, about 25π rad/s,about 30π rad/s, about 35π rad/s, about 40π rad/s, about 45π rad/s,about 50π rad/s, about 55π rad/s, about 60π rad/s, about 65π rad/s,about 70π rad/s, about 75π rad/s, about 80π rad/s, about 85π rad/s,about 90π rad/s, about 95π rad/s, about 100π rad/s, about 105π rad/s,about 110π rad/s, and about 115π rad/s. In still further aspects, the ACcan have any β values and any frequency values between any two foregoingvalues.

In yet further aspects, the AC can have the predetermined β values fromgreater than 0 to about 1, about 0.1, including exemplary values ofabout 0.2, about 0.3, about 0.4, about 0.5, about 0.6, about 0.7, about0.8, and about 0.9; and the predetermined frequency values from greaterthan 0 to about 60 Hz, including exemplary values of about 0.1 Hz, about0.5 Hz, about 1 Hz, about 2 Hz, about 5 Hz, about 10 Hz, about 20 Hz,about 30 Hz, about 40 Hz, and about 50 Hz. In still further aspects, theAC can have any β values and any frequency values between any twoforegoing values.

It is understood that in some aspects, the systems disclosed herein aremore energy efficient when is compared to the common systems without thepresence of the AC. In certain aspects, the inventive systems exhibit anincrease in the energy efficiency by at least about 5%, at least about7%, at least about 10%, at least about 12%, or at least about 14% whencompared to the substantially identical reference system in the absenceof the AC. In yet other aspects, the electrolytic cell present in thedisclosed system can exhibit an increase in the energy efficiency thatis substantially proportional to a reduction in the anode-to-cathodedistance.

EXAMPLES

The following examples are put forth so as to provide those of ordinaryskill in the art with a complete disclosure and description of how thecompounds, compositions, articles, devices and/or methods claimed hereinare made and evaluated and are intended to be purely exemplary and arenot intended to limit the disclosure. Efforts have been made to ensureaccuracy with respect to numbers (e.g., amounts, temperature, etc.), butsome errors and deviations should be accounted for.

Example 1

The stability of a simulated mechanical analog to the fluid layersinside the H-H cell, as shown in FIG. 1 , has been studied.

Since the liquid layers (e.g., aluminum and electrolyte) are broad andshallow, a slight tilting of the interface substantially redistributesthe current density J in the cell. Furthermore, the conductivity ofaluminum is much higher than that of the carbon electrodes, andtherefore, excess current is drawn into the aluminum in regions wherethe thickness of the electrolyte is reduced, and less current is drawnat regions where the thickness of the electrolyte is increased. In otherwords, the resulting perturbation current density j is downwards at thewave crests and upwards at the troughs.

The perturbed current density j produces a perturbed magnetic fielddenoted by b. J₀ and B₀ have been chosen to define the unperturbedcurrent density and magnetic field. Then, the perturbed Lorentz forcecan be defined by Eq. 1:

J ₀ ×b+j×B ₀  (Eq. 1)

It was previously shown that the interface can support an infinitenumber of conventional standing waves in the absence of a magneticfield. The normal modes associated with these gravitational standingwaves form an orthogonal set of functions. Hence, an arbitrarydisturbance can be represented as a superposition of many gravitationalmodes. The redistribution of current caused by one gravitational modegives rise to a perturbed Lorentz force, which can excite many othergravitational modes coupling certain modes. This coupling leads toinstability involving two or more adjacent gravitational frequencies

The mechanical model described herein includes a sinusoidal componentthat is introduced to the unperturbed current density. The model 400(FIG. 4 ) considered herein has a compound pendulum that is made from aflat aluminum plate 404 attached to a parallel fixed surface by apivoted rigid struct that allows the plate to freely swing about twohorizontal axes (x and y). The origin of the system (O) is located atthe connection between the aluminum plate and the strut. The initialcryolite layer (402) height is denoted by h₀, and the aluminum height byH. The aluminum plate has horizontal dimensions L_(x) and L_(y).

A uniform current density is defined by Eq. 2 is applied to the system

J=J ₀(1+β sin(ω_(b) t+ϕ))  (Eq. 2)

where J₀ is the DC, β is a non-dimensional amplitude ratio of the ACcompared to that of the DC, ω_(b) is the angular frequency of the AC,and ϕ is its phase.

A uniform vertical magnetic field, B_(z), is imposed. As describedherein and shown in FIG. 4 , ρ and σ represent the density andelectrical conductivity, respectively. The origin of the axes lies atthe center of the electrolyte-plate interface in the unperturbed state(marked with an “O”).

Example 2

To simplify the calculations, the following assumptions have beenintroduced. First, the characteristic time scale for the wave motion(period of oscillation) was found to be much greater than the magneticfield diffusion time. In such aspects, it was assumed that the currentwould immediately relax to a new equilibrium distribution each time theinterface moves. Further, it was assumed that j feeding into thealuminum does not penetrate the carbon cathode block. Without wishing tobe bound by any theory, it was assumed that this is due to theelectrical resistivity of carbon is much higher than that of aluminum.It was further assumed that the fluid is inviscid. In such an aspect, itwas assumed that the damping of high-wavenumber perturbations would notbe mimicked. It was further assumed that surface tension could beignored, and there is no background motion in the unperturbed state(u₀=0). This imposes a limitation on B₀ as it must satisfy Eq. 3 toensure that the perturbations are about an equilibrium configuration:

∇×(J ₀ ×B ₀)=0  (Eq. 3)

Further, the shallow water approximation was used, kh<<1, where k is atypical wavenumber. Such an assumption leads to the followingsimplifications: i) the perturbed current j is vertical in theelectrolyte (due to the thin layer of electrolyte having the dominantresistance to the current flow forcing the current to pass directlydownwards through this layer); ii) aluminum is a great conductor andassumed to be an equipotential surface (j is horizontal in the aluminumand is uniformly distributed across the plate. In other words, theperturbed current “shorts” through the aluminum); iii) the aspect ratio,kh<<1 can give rise to the fact that the perturbed current in theelectrolyte is much smaller than that in the aluminum, i.e.,j_(c)<<j_(Al) (hence, the perturbed Lorentz force acting on theelectrolyte can be neglected); iv) the velocity in each layer is uniformin a z axis and a horizontal axis (it follows from the fact that theLorentz force in the aluminum plate is independent of depth); and v)when “H” and “V” denote horizontal and vertical components respectively,j_(H)×B_(H) is vertical, and thus perturbs the vertical pressuregradient only and can be neglected. J_(V)×B_(H) is much smaller thanj_(H)×B_(V) because of (ii). Also, j_(V)×B_(V) is of order kH smallerthan j_(V)×B_(H). Therefore, the dominant contribution to the perturbedLorentz force in the aluminum can be presented according to Eq. 4:

j _(V) ×B _(H) or j×(B _(z) ê _(z))  (Eq. 4)

Further, it was assumed that the aluminum plate and electrolyte layerare thin and broad, i.e., L_(x), L_(y)>>h, H. It was also assumed, forconvenience, that B_(z)>>B_(x), B_(y); and thatρ_(electrolyte)<<ρ_(aluminium), which implies that the inertia of theelectrolyte can be ignored.

Example 3 Derivation of Equations of Motion Example 3.1 BoundaryConditions

For the following sections, subscripts “e” and “al” are used to denotethe electrolyte and the aluminum plate, respectively. In addition,superscripts are used to denote the axis at which the system property orcharacteristic is considered. For example, j_(e) ^(z) refers to theperturbed current density in the electrolyte along the z-direction.

Since aluminum is much more electrically conductive than the electrolyteand the carbon cathode block, it can be assumed that the currentperturbations form closed loops inside of the aluminium plate (O.Zikanov. Metal pad instabilities in liquid metal batteries. PHYSICALREVIEW E, 92(6), 2015). This assumption translates mathematically to thefollowing boundary conditions (Eqs. 5-7):

j _(al) ^(x) ·{circumflex over (n)}| _(c)=0=j _(al) ^(y) ·{circumflexover (n)}| _(c); where c indicates“boundary of plate”  (Eq. 5)

j _(al) ^(z)|_(z=0) =j _(e) ^(z)  (Eq. 6)

j _(al) ^(z)|_(z=−H)=0  (Eq. 7);

where {circumflex over (n)} is a unit normal vector to the plate'sboundary in the x-y plane.

Example 3.2 Change in Electrolyte Thickness Under x and y Rotations

FIG. 4 shows an exemplary model used herein. The following parametershave been defined: h(x, y) is assumed to be the local thickness of theelectrolyte, and h₀ is assumed to be the equilibrium value of h. It wasfurther assumed that a small rotation of θ_(x) of the plate about thex-axis is present. Further, Δh_(x) has been assumed to be theperpendicular distance from the top of the plate to the y-axis (seeFIGS. 5A-5C).

Using simple geometrical arguments, it was shown that the angle betweenthe aluminum plate and the y-axis is θ_(x). Thus, according to Eq. 8

$\begin{matrix}{{\tan\theta_{x}} = {\frac{\Delta h_{x}}{y - {h_{0}\sin\theta_{x}}}.}} & \left( {{Eq}.8} \right)\end{matrix}$

However, for small θ_(x), tan θ_(x)≈sin θ_(x)=θ_(x). Substituting thisassumption into Eq. 8, it can be found that:

Δh _(x) ≈yθ _(x) −h ₀θ_(x) ²  (Eq. 9)

Similarly, if a small rotation of θ_(y) of the plate about the y-axis isassumed, then Δh_(y) is set to be the perpendicular distance between thetop of the plate and the x-axis. Then, using the small-angleapproximation (Eq. 10):

Δh _(y) ≈xθ _(y) +h ₀θ_(y) ²  (Eq. 10)

In such instances, a rotation of θ_(x) decreases the electrolytethickness by Δh_(x), while a rotation of θ_(y) increases the electrolytethickness by Δh_(y). Using the principle of superposition, for acombined rotation of θ_(x) and θ_(y), the electrolyte thickness is givenby (Eq. 11):

$\begin{matrix}\begin{matrix}{{h\left( {x,y} \right)} = {h_{0} + {\Delta h_{y}} - {\Delta h_{x}}}} \\{\approx {h_{0} + {x\theta_{x}} + {h_{0}\theta_{y}^{2}} - {y\theta_{x}} + {h_{0}\theta_{x}^{2}}}} \\{\approx {h_{0} + {x\theta_{x}} - {y\theta_{x}}}}\end{matrix} & (11)\end{matrix}$

3.3 Perturbed Current Density in the Electrolyte

Since the electrical conductivity of the aluminum plate is much higherthan that of the electrolyte, the aluminum plate is an equipotentialsurface at ϕ₀. Since the infinitesimal perturbations of the interfaceare much smaller than the thickness of the electrolyte and the thicknessof the plate, it is assumed that ϕ₀ does not change when the interfaceis tilted. Using the narrow gap approximation, Φ can be found (Eq. 12):

$\begin{matrix}{\Phi = {\frac{\Phi_{0^{z}}}{h} = \frac{\Phi_{0^{z}}}{h_{0} + {x\theta_{y}} - {y\theta_{x}}}}} & \left( {{Eq}.12} \right)\end{matrix}$

Here, J_(e) denotes the total current density in the electrolyte. Then,J_(e)=J₀+j_(e), where J₀=J₀(1+β sin(w_(b)t))(−ê_(z)) is the unperturbedcurrent density, and j_(e) is the perturbed current density in theelectrolyte. Here, based on previously presented assumptions, j_(e) ispurely vertical, and therefore, J_(e) must be purely vertical.Therefore, as shown in Eq. 13:

$\begin{matrix}{J_{e} = {{{- \sigma}{\nabla{\Phi\left( {- {\hat{e}}_{z}} \right)}}} = {{{- \sigma}\frac{\partial\Phi}{\partial z}\left( {- {\hat{e}}_{z}} \right)} = {{- \sigma}\frac{\Phi_{0}}{h_{0} + {x\theta_{y}} - {y\theta_{x}}}\left( {- {\hat{e}}_{z}} \right)}}}} & \left( {{Eq}.13} \right)\end{matrix}$

When θ_(x)=0=θ_(y), the system is at equilibrium, and there is noperturbed current. Hence, as shown in Eq. 14

$\begin{matrix}{{J_{e}❘_{\theta_{x} = {0 = {\theta y}}}} = {{{- \frac{\sigma\Phi_{0}}{h_{0}}}\left( {- {\hat{e}}_{z}} \right)} = {J_{0} = {{J_{0}\left( {1 + {\beta{\sin\left( {w_{b}t} \right)}}} \right)}\left( {- {\hat{e}}_{z}} \right)}}}} & \left( {{Eq}.14} \right)\end{matrix}$

Using Eq. 14, Eq. 13 can be written as Eq. 15:

$\begin{matrix}{J_{e} = {\frac{{J_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}{h \cdot 0}}{h_{0} + {x\theta_{y}} - {y\theta_{x}}}\left( {- {\hat{e}}_{z}} \right)}} & \left( {{Eq}.15} \right)\end{matrix}$

Using Eq. 14 and Eq. 15, the perturbed current density in theelectrolyte can be described as shown in Eq. 16:

$\begin{matrix}\begin{matrix}{j_{e} = {J_{e} - J_{0}}} \\{= {\left( {{- \frac{{J_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}h_{0}}{h_{0} + {x\theta_{y}} - {y^{\theta}x}}} + {J_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}} \right)\left( {\hat{e}}_{z} \right)}} \\{= {{J_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}\left( \frac{{x\theta_{y}} - {y\theta_{x}}}{h_{0} + {x\theta_{y}} - {y\theta_{x}}} \right)\left( {\hat{e}}_{z} \right)}}\end{matrix} & \left( {{Eq}.16} \right)\end{matrix}$

If γ=xθ_(y)−yθ_(x), then

$j_{e} = {\frac{\gamma}{h_{o} + \gamma}{\left( {\hat{e}}_{z} \right).}}$

Taylor expanding

$\frac{\gamma}{h_{o} + \gamma}$

around γ=0 gives Eq. 17:

$\begin{matrix}{{{{{\frac{\gamma}{h_{0} + \gamma} \approx {0 + \frac{h_{0} + \gamma - \gamma}{\left( {h_{0} + \gamma} \right)^{2}}}}❘}_{\gamma = 0}\left( {\gamma - 0} \right)} + \ldots} \approx \frac{\gamma}{h_{0}}} & \left( {{Eq}.17} \right)\end{matrix}$

Therefore, the perturbed current density in the electrolyte is (Eq. 18):

$\begin{matrix}\begin{matrix}{j_{e} = {{J_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}\frac{\gamma}{h_{0} + \gamma}\left( {\hat{e}}_{z} \right)}} \\{\approx {\frac{{J_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}\gamma}{h_{0}}\left( {\hat{e}}_{z} \right)}} \\{= {\frac{{J_{0}\left( {\beta{\sin\left( {\omega_{b}t} \right)}} \right)}\left( {{x\theta_{y}} - {y\theta_{x}}} \right)}{h_{0}}\left( {\hat{e}}_{z} \right)}}\end{matrix} & {{Eq}.18}\end{matrix}$

3.4 Aluminum Plate's Moment of Inertia about x and y-Axis Through Pivot

In this example, and shown in FIGS. 7-8 , COM is assumed to be a centerof mass, and I_(x)=∫^(M)r²dM; where r is a perpendicular distance from apoint to the “x”-axis.

Moment of inertia about axis is parallel to the x-axis through COM ofthe plate (x′ axis), as shown in FIGS. 7-8 . The x′ axis is at z=−H/2and y=0, resulting in the COM having coordinates (0, 0, −H/2).Considering that a point M has coordinates (x, y, z) at the aluminumplate, then the orthogonal project of M on the x′ axis has coordinates(x, 0, −H/2).

$\begin{matrix}{r^{2} = {{\left( {x - x} \right)^{2} + \left( {y - 0} \right)^{2} + \left( {z + \frac{H}{2}} \right)^{2}} = {y^{2} + \left( {z + \frac{H}{2}} \right)^{2}}}} & \left( {{Eq}.19} \right)\end{matrix}$ $\begin{matrix}{{I_{{x^{\prime}}_{COM}} = {\int^{M}{r^{2}dm}}};{{{where}dm} = {{\rho_{a}dV} = {{\rho_{a}L_{x}{dzdy}} = {\int_{{- H}/2}^{0}{\int_{{- L_{y/}}/2}^{L_{y}/2}\left\lbrack {{y^{2} + {\left( {z + \frac{H}{2}} \right)^{2}\rho_{a}L_{x}{dydz}}} = {{\rho_{a}L_{x}{\int_{- H}^{0}{\left\lbrack {\frac{L_{y}^{3}}{12} + {\left( {z + \frac{H}{2}} \right)^{2}L_{y}}} \right\rbrack{dz}}}} = {{\rho_{a}{L_{x}\left\lbrack {\frac{HL_{y}^{3}}{12} + \frac{H^{3}L_{y}}{3} - \frac{H^{3}L_{y}}{2} + \frac{H^{3}L_{y}}{4}} \right\rbrack}} = {\rho_{a}L_{x}H{L_{y}\left\lbrack {\frac{L_{y}^{2}}{12} + \frac{H^{2}}{12}} \right\rbrack}\text{ }}}}} \right.}}}}}} & \left( {{Eq}.20} \right)\end{matrix}$ $\begin{matrix}{{Similarly},{I_{{y^{\prime}}_{COM}} = {\rho_{a}L_{x}H{L_{y}\left\lbrack {\frac{L_{x}^{2}}{12} + \frac{H^{2}}{12}} \right\rbrack}}}} & \left( {{Eq}.21} \right)\end{matrix}$

3.5 Moment of Inertia about Axes Parallel to x′ and y′ Passing ThroughPivot

Based on the model schematic shown in FIG. 9 :

$\begin{matrix}{{{I_{{x^{''}}_{COM}} = {I_{{x^{\prime}}_{COM}} + {md}^{2}}};{m = {\rho_{a}L_{x}{HL}_{y}{and}}}}{d^{2} = \left( {h_{0} + \frac{H}{2}} \right)^{2}}} & \left( {{Eq}.22} \right)\end{matrix}$

where m is a mass of plate, and dis the distance between two axes.

Hence,

$\begin{matrix}{I_{x^{''}} = {\rho_{a}L_{x}{{HL}_{y}\left\lbrack {\frac{L_{y}^{2}}{12} + \frac{H^{2}}{12} + \left( {h_{0} + \frac{H}{2}} \right)^{2}} \right\rbrack}}} & \left( {{Eq}.23} \right)\end{matrix}$ Similarly, $\begin{matrix}{{I_{y^{''}} = {\rho_{a}L_{x}{{HL}_{y}\left\lbrack {\frac{L_{x}^{2}}{12} + \frac{H^{2}}{12} + \left( {h_{0} + \frac{H}{2}} \right)^{2}} \right\rbrack}}};} & \left( {{Eq}.24} \right)\end{matrix}$

where y″ is parallel to y′ passing through pivot.

3.6 Torques Due to Perturbed Lorentz Force

The horizontal components of the perturbed current within the aluminumplate interact with the unperturbed vertical magnetic field,B₀=B_(z)(ê_(z)), giving rise to a perturbed Lorentz force:

f=j _(al) ×B ₀  (Eq. 25)

Where f is the Lorentz force per unit volume, Eq. 25 can be written as:

f ^(x) =j _(al) ^(y) B ₀(ê _(x))  (Eq. 26)

f ^(y) =j _(al) ^(x) B ₀(−ê _(y))  (Eq. 27)

If both sides of Eq. 26 are integrated along y and z directions, and Eq.27 along z and x directions, then:

∫_(−L) _(y) _(/2) ^(L) ^(y) ^(/2)∫_(−H) ⁰ f ^(x) dzdy=∫ _(−L) _(y) _(/2)^(L) ^(y) ^(/2)∫_(−H) ⁰ j _(al) ^(y) B ₀ dzdy(ê _(x))  (Eq. 28)

∫_(−L) _(x) _(/2) ^(L) ^(x) ^(/2)∫_(−H) ⁰ f ^(y) dzdx=∫ _(−L) _(x) _(/2)^(L) ^(x) ^(/2)∫_(−H) ⁰ j _(al) ^(x) B ₀ dzdx(−ê _(y))  (Eq. 29)

The right-hand side of Eq. 28 and Eq. 29 are, respectively, the net flowof perturbed current along the x and y-directions inside of the aluminumplate. Hence, the distribution of the Lorentz force components along thex and y-directions are respectively:

$\begin{matrix}{{F^{x}(y)} = {{{I_{al}^{y}(y)}{B_{0}\left( {\hat{e}}_{x} \right)}} = {{- {\frac{J_{0}{B_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}\left( {\theta_{x}L_{x}} \right)}{2h_{0}}\left\lbrack {\left( \frac{L_{y}}{2} \right)^{2} - y^{2}} \right\rbrack}}\left( {\hat{e}}_{x} \right)}}} & \left( {{Eq}.30} \right)\end{matrix}$ $\begin{matrix}{{F^{y}(x)} = {{{- {I_{al}^{x}(x)}}{B_{0}\left( {\hat{e}}_{y} \right)}} = {{- {\frac{J_{0}{B_{0}\left( {1 + {\beta\sin\left( {\omega_{b}t} \right)}} \right)}\left( {\theta_{y}L_{y}} \right)}{2h_{0}}\left\lbrack {\left( \frac{L_{x}}{2} \right)^{2} - x^{2}} \right\rbrack}}\left( {\hat{e}}_{y} \right)}}} & \left( {{Eq}.31} \right)\end{matrix}$

Using Eq. 30 and Eq. 31 and referring to FIGS. 6A-6B, the distributionof the torque (arising from the Lorentz force) components, about thepivot, along the x and y-directions are, respectively, found to be:

$\begin{matrix}{{\tau^{x}(x)} = {{r_{\bot}^{x}{F^{y}(x)}\left( {{- {\hat{e}}_{z}} \times {\hat{e}}_{y}} \right)} = {\left( {h_{0} + \frac{H}{2}} \right){F^{y}(x)}\left( {\hat{e}}_{x} \right)}}} & \left( {{Eq}.32} \right)\end{matrix}$ $\begin{matrix}{{\tau^{y}(y)} = {{r_{\bot}^{y}{F^{x}(y)}\left( {{- {\hat{e}}_{z}} \times {\hat{e}}_{x}} \right)} = {\left( {h_{0} + \frac{H}{2}} \right){F^{x}(y)}\left( {- {\hat{e}}_{y}} \right)}}} & \left( {{Eq}.33} \right)\end{matrix}$

Finally, the net torques along the x and y-direction were respectivelyobtained by integrating Eq. 32 along the x-direction and Eq. 33 alongthe y-direction:

$\begin{matrix}{\tau_{Net}^{x} = {{\int_{- \frac{L_{x}}{2}}^{\frac{L_{x}}{2}}{{\tau^{x}(x)}{dx}}} = {{- \left( {h_{0} + \frac{H}{2}} \right)}\frac{J_{0}{B_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}\left( {\theta_{y}L_{y}} \right)}{h_{0}}\left( \frac{L_{x}}{12} \right)^{3}}}} & \left( {{Eq}.34} \right)\end{matrix}$ $\begin{matrix}{\tau_{Net}^{y} = {{\int_{- \frac{L_{y}}{2}}^{\frac{L_{y}}{2}}{{\tau^{y}(y)}{dy}}} = {\left( {h_{0} + \frac{H}{2}} \right)\frac{J_{0}{B_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}\left( {\theta_{x}L_{x}} \right)}{h_{0}}\left( \frac{L_{y}}{12} \right)^{3}\left( {\hat{e}}_{y} \right)}}} & \left( {{Eq}.35} \right)\end{matrix}$

3.7 Equations of Motion

Conservation of angular momentum around horizontal axes parallel to xand y axes when the aluminum plate is at θ_(x)=0=θ_(y), and passingthrough the pivot is used to describe the motion of the plate. As shownin FIGS. 6A-6B, the only torques acting on the plate are the ones due tothe Lorentz and gravity forces.

I _(xx)α_(x)=Στ=τ_(Net) ^(x)+τ_(gravity) ^(x)  (Eq. 36)

I _(yy)α_(y)=Στ=τ_(Net) ^(y)+τ_(gravity) ^(y)  (Eq. 37)

Where:

${I_{xx} = {\rho_{a}L_{x}{{HL}_{y}\left\lbrack {\frac{L_{y}^{2}}{12} + \frac{H^{2}}{12} + \left( {h_{0} + \frac{H}{2}} \right)^{2}} \right\rbrack}{and}}}{I_{yy} = {\rho_{a}L_{x}{{HL}_{y}\left\lbrack {\frac{L_{x}^{2}}{12} + \frac{H^{2}}{12} + \left( {h_{0} + \frac{H}{2}} \right)^{2}} \right\rbrack}}}$

are the moments of inertia (as shown above), α_(x)={umlaut over(θ)}_(x)ê_(x) and α_(y)={umlaut over (θ)}_(y)ê_(y) are the angularacceleration along the x and y direction, respectively. Whileτ_(gravity) ^(x) and τ_(gravity) ^(y) are the torques due to thegravitational force along the x and y directions. Referring to FIGS.6A-6B:

$\begin{matrix}{\tau_{gravity}^{x} = {{r_{\bot} \times {mg}} = {\rho_{a}L_{x}L_{y}{{Hg}\left( {h_{0} + \frac{H}{2}} \right)}{\theta_{x}\left( {- {\hat{e}}_{x}} \right)}}}} & \left( {{Eq}.38} \right)\end{matrix}$ $\begin{matrix}{\tau_{gravity}^{y} = {{r_{\bot} \times {mg}} = {\rho_{a}L_{x}L_{y}{{Hg}\left( {h_{0} + \frac{H}{2}} \right)}{\theta_{y}\left( {- {\hat{e}}_{y}} \right)}}}} & \left( {{Eq}.39} \right)\end{matrix}$

The Eq. 38 and 39 can be rewritten as:

$\begin{matrix}{{{\overset{¨}{\gamma}}_{x} + {\omega_{x}^{2}\gamma_{x}}} = {{- \frac{J_{0}{B_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}}{\rho_{al}H}}\gamma_{y}}} & \left( {{Eq}.40} \right)\end{matrix}$ $\begin{matrix}{{{\overset{¨}{\gamma}}_{y} + {\omega_{y}^{2}\gamma_{y}}} = {\frac{J_{0}{B_{0}\left( {1 + {\beta{\sin\left( {\omega_{b}t} \right)}}} \right)}}{\rho_{al}H}\gamma_{x}}} & \left( {{Eq}.41} \right)\end{matrix}$

where

${\gamma_{x} = {{\frac{\theta_{x}}{L_{x}^{2}}{and}\gamma_{y}} = \frac{\theta_{y}}{L_{y}^{2}}}};$

where θ_(x), θ_(y) rotations about the x and y-axis, respectively.

In the cases where ϕ is not zero, the Eqs. 40 and 41 can also include aphase component:

$\begin{matrix}{{{\overset{¨}{\gamma}}_{x} + {\omega_{x}^{2}\gamma_{x}}} = {{- \frac{J_{0}{B_{0}\left( {1 + {\beta{\sin\left( {{\omega_{b}t} + \phi} \right)}}} \right)}}{\rho_{al}H}}\gamma_{y}{and}}} & \left( {{Eq}.42} \right)\end{matrix}$ $\begin{matrix}{{{\overset{¨}{\gamma}}_{y} + {\omega_{y}^{2}\gamma_{y}}} = {\frac{J_{0}{B_{0}\left( {1 + {\beta{\sin\left( {{\omega_{b}t} + \phi} \right)}}} \right)}}{\rho_{al}H}\gamma_{x}}} & \left( {{Eq}.43} \right)\end{matrix}$

The double dot ({umlaut over (γ)}_(x)) represents the second derivativewith respect to time, specifically {umlaut over (γ)}_(x) represents anangular acceleration in the x-direction. The natural frequencies of thepure gravitational oscillations are denoted by:

$\begin{matrix}{{\omega_{x}^{2} \approx \frac{g\left( {h_{0} + \frac{H}{\left. 2 \right)}} \right.}{L_{y}^{2}/12}},{and}} & \left( {{Eq}.44} \right)\end{matrix}$ $\begin{matrix}{{\omega_{y}^{2} \approx \frac{g\left( {h_{0} + \frac{H}{\left. 2 \right)}} \right.}{\frac{L_{x}^{2}}{12}}};} & \left( {{Eq}.45} \right)\end{matrix}$

where g is 9.81 m/s² is the acceleration due to gravity.

Example 4 4.1 Case Study I: Uniform Current Density, β=0

MATLAB, more specifically “ode45”, was utilized to solve the system ofcoupled ODEs representing the Equations of motion of the pendulum (Eqs.40 and 41). The extreme case where there is no sinusoidal component tothe unperturbed current (β=0) was considered first.

It was shown (P. A. Davidson et al., Stability of interfacial waves inaluminium reduction cells. Journal of Fluid Mechanics, 362:273-295,1998) that the onset of instability occurs when:

$\begin{matrix}{{❘{\omega_{x}^{2} - \omega_{y}^{2}}❘} = {\frac{2J_{0}B_{0}}{\rho_{al}H} = {2a}}} & \left( {{Eq}.46} \right)\end{matrix}$

The oscillations are stable as long as |ω_(x) ²−ω_(y) ²|>2a. For celldimensions of h₀=4.5 cm, H=0.2 m, L_(x)=11 m and L_(y)=2.7 m, thenatural frequencies are ω_(x)=1.5302 Hz and ω_(y)=0.3756 Hz. Thisimplies that the stability threshold is at a=1.100206. It was shown thatfor the case of a DC current only, the oscillations in the model areunstable for values of “a” higher than the critical threshold. FIGS.10A-10D show numerical results for different values of a. It can be seenthat the system becomes unstable at a=1.100207.

FIGS. 11A-11C show additional exemplary data showing oscillations thatcan be present in the aluminum pot model with only DC applied. Thenatural frequencies of the model are ω_(x)=1.5302 Hz and w_(y)=0.3756Hz. Initially, at t=0 seconds, a small wave is applied. As shown in FIG.11A, at the amplitude of the oscillations for 60 seconds and thestability parameter “a” slightly below the critical value, theoscillations are finite and growing in amplitude slowly. FIG. 11B showsthe same oscillations in panel a but for a substantially longer time.The oscillations are clearly finite, which indicates stability. As shownin FIG. 11C, when the stability parameter “a” is increased slightlyabove the critical value, the oscillation amplitude increases by fiveorders of magnitude. Without wishing to be bound by any theory, thisrapid growth of oscillations is associated with instability.

4.2 Case Study II: β≠0

With the numerical solver agreeing with theoretical results, attemptingto stabilize the system by varying β and ω_(b) came next. First,a=1.1003 was set to ensure that the system is unstable, and ω_(b)=2π wasfixed. β varied from 0.1 to 0.5 in steps of 0.05. A stable state wasachieved at β≥0.3, as seen in FIGS. 12A-12I.

FIGS. 13A-13C show other exemplary oscillations of the aluminum potmodel with the stability parameter a=1.1354. This value of a representsa 10% decrease in the cryolite thickness in the model having

${\omega_{b} = {\frac{\pi}{2}{rad}/s}},{\phi = {0{rad}}},{\omega_{x} = {1.5302{Hz}}},$

and ω_(y)=0.3756 Hz. Initially, at t=0 seconds, a small wave is applied.As shown in FIG. 13A, β=0, which corresponds to DC current only. Asshown, the oscillations are rapidly growing, indicating instability. InFIG. 13B, β=0.23, meaning that an AC current is added with an amplitudeof 23% of the DC current. At these β values, the oscillations are stillappeared to be unstable. FIG. 13 C shows results for the case withβ=0.24. It can be seen that at this β value, oscillations do not grow.This shows the aluminum pot is stabilized by adding an AC current with asuitable amplitude.

Next, β=0.1 and a=1.1003 were fixed creating an unstable scenario. Inthis experiment, ω_(b) was varied from 0 to 2π in steps of π/4. Unlikethe case with varying β, the stability was not observed above a certainthreshold. Instead, it was observed that the system stabilized atdiscrete values of ω_(b). As shown in FIGS. 14A-14I stability wasachieved at ω=π/4, π/2, π, 5π/4.

FIGS. 15A-15C show additional exemplary oscillations of the aluminum potmodel with the stability parameter a=1.1354. This value of a representsa 10% decrease in the cryolite thickness in the model. The followingsystem parameters have been assumed: β=0.23, ϕ=0 rad, ω_(x)=1.5302 Hz,and ω_(y)=0.3756 Hz. Initially, at t=0 seconds, a small wave is applied.FIG. 15A show results for ω_(b)=0 rad/s, which corresponds to DC currentonly. As shown, the oscillations are rapidly growing, indicatinginstability. As shown in FIG. 15B ω_(b)=0.3 rad/s, and the oscillationsare still unstable. As shown in FIG. 15C at ω_(b)=0.5 rad/s theoscillations' amplitude is not growing anymore. This shows again thataluminum pot can be stabilized by adding an AC current with a suitableangular frequency.

FIG. 16 depicts a schematic of the stability of the oscillations in thealuminum pot model for different combinations of β and ω_(b). Thestability parameter a=1.1354, which represents a 10% decrease in thecryolite thickness in the model. ϕ=0 rad, ω_(x)=1.5302 Hz, andω_(y)=0.3756 Hz. As shown, for β=0 or ω_(b)=0 (which indicates a DCcurrent only), the oscillations are always unstable when a is above itscritical value (cryolite thickness is below its critical values). Forsome combinations of β and ω_(b) the oscillations can be successfullystabilized.

Example 5

In this example, the shape of the interface was simulated using anMHD-VALDIS, a software package written specifically for simulatingaluminum reduction cells and used widely throughout the industry. Theinstantaneous and time-averaged interface shapes are shown in FIG. 17 ,along with the difference between the two, as referred herein as adeviation. This deviation was used to quantify the interfacedeformation.

Without wishing to be bound by any theory, it was assumed that theresonant instability can occur when two gravity-wave modes on thealuminum-salt interface couple to each other, producing a wave thatcirculates counter-clockwise and is amplified by the electrical current.FIG. 18A shows the evolution of the interface deformation, illustratinga circulating motion. In this simulation, an anode-cathode distance(ACD, the thickness of the salt layer) was assumed to be 4.0 cm, and avalue of a direct current was assumed to be 180 kA. As shown in FIG.18E, under such simulated conditions, the resonant instability hasoccurred, and the root-mean-square deformation increased dramatically.This behavior was consistent with an exponential increase over time, andthe best-fit exponential curve had a growth rate of 0.002915 Hz,corresponding to a doubling time of 321 s. An aluminum reduction cell isnot expected to be operable under these conditions.

At any moment, the shape of the interface deviation can be written as asum of gravity wave modes, as shown in Eq. 47:

$\begin{matrix}{{z\left( {x,y,y} \right)} = {\sum_{n,{m = 0}}^{\infty}{{C_{nm}(t)}\cos\frac{n\pi}{L_{x}}x\cos\frac{m\pi}{Ly}y}}} & (47)\end{matrix}$

where z is the deviation of the interface from its average shape; (x; y)are Cartesian coordinates in the horizontal plane; t is time; (n;m) arepositive integers; the spectral power C_(nm)(t), which varies over time,is the magnitude of the mode specified by (n;m); and (L_(x); L_(y)) arethe lengths of the cell in the (x; y) directions, respectively. FIG. 18Fshows that the strongest modes in this simulation are (n;m)=(2; 0) and(0; 2). FIG. 18G shows the time evolution of their strengths. As bothgrow, they oscillate 180° out of phase (one is strong while the other isweak), the hallmark of a circulating wave.

In aspects where no current is present, the frequency of any wave mode(n, m) can be calculated according to Eq. 48:

$\begin{matrix}{f_{nm} = {\frac{1}{2\pi}\sqrt{\frac{\rho_{1} - \rho_{2}}{\frac{\rho_{1}}{h_{1}} + \frac{\rho_{2}}{h_{2}}}{g\left( {\frac{n^{2}\pi^{2}}{L_{x}^{2}} + \frac{m^{2}\pi^{2}}{L_{y}^{2}}} \right)}}}} & (48)\end{matrix}$

Where (ρ₁, ρ₂) are the densities of the aluminum and salt, respectively;(h₁; h₂) are the thicknesses of the aluminum and salt, respectively; andg is the gravitational acceleration. This expression can be used toaccurately approximate the frequencies of wave modes in simulation, eventhough a large current is present, as shown in FIG. 18H. The plot showsthe power spectrum of oscillation of a single point on the interface.Its motion is dominated by a frequency that closely matches thetheoretical frequency of the (2; 0) mode and is almost exactly half thetheoretical frequency of the (0; 2) mode, characteristic of a resonance.

Without wishing to be bound by any theory, it is hypothesized that theresonant instability occurs only when two or more adjacent gravity-wavemodes couple to produce a counter-clockwise circulation. The methodsdisclosed herein prevent such a formation and growth of such waves byadding an oscillating component to the electrical current runningthrough the cell. It is understood that any frequencies and amplitudesas disclosed herein can be chosen for the oscillating current component.

In certain aspects, the frequencies are chosen for an oscillatingcomponent such that a standing wave is formed instead of circulation.FIGS. 19A-D show the evolution of the interface deformation in asimulation with the same conditions as above (4.0 cm ACD and 180 kAsteady current) but with an addition of a 19.8 kA oscillating componentwith a frequency of 0.045 Hz. It was found that the interface shape insuch aspect evolves not according to circulation but according to astanding wave. In such aspects, methods disclosed herein prevent theinterface deformation from growing large over time, as shown in FIG.19E. It is further understood that under such conditions, an aluminumreduction cell can operate successfully.

FIG. 19F shows that the strongest modes in this simulation is (n;m)=(0;2). The (2; 0) mode is present but is much weaker than in the simulationwithout an oscillating current component. FIG. 19G shows the timeevolution of mode strengths. Instead of growing stronger and strongerover time, the modes oscillate steadily. The (2; 0) mode was found toremain weak throughout the simulation. FIG. 19H shows the power spectrumof oscillation of a single point on the interface. Its motion isdominated by two frequencies, one closely matching the theoreticalfrequency of the (0; 2) mode and the other closely matching thetheoretical frequencies of the (2; 0) and (0; 1) modes. The ratio of thetwo frequencies is almost exactly two, characteristic of a resonance.

Altogether, these results show that adding an oscillating currentcomponent with frequency, for example, and without limitation of 0.045Hz, can drive an (n;m)=(0; 2) mode whose frequency is nearly the same asthe 0.0498 Hz predicted by theory. The (0; 2) mode can also prevent theformation and growth of the (2; 0) and (0; 1) modes that would result ina circulating wave and give rise to a resonant instability, making thealuminum cell inoperable. Instead, standing waveforms and the cell canbe operated indefinitely.

Other simulations were performed using a smaller value of ACD. In oneexemplary simulation, an ACD of 3.8 cm and 180 kA steady current with a19.8 kA oscillation at 0.045 Hz were used. The (0; 2) mode was againexcited, and the cell was again stable, allowing indefinite operation.

This cell is known to be stable without an oscillating component if theACD is at least 4.2 cm. The methods disclosed herein allow, for example,a 9.5% reduction in ACD, which would cause a similar reduction in energyconsumption and substantial cost savings.

In an additional exemplary simulation, an ACD of 4.0 cm and 180 kAsteady current with a 19.8 kA oscillation at 0.069 Hz were also tested.That frequency excited the (6; 0) mode, whose frequency is predicted tobe 0.0688 Hz. The (6; 0) mode was found to couple with the (0; 2) modeto produce a standing wave, preventing the (0; 1) and (2; 0) modes fromcoupling to produce circulation and the resulting resonant instability.

In an additional exemplary simulation, an ACD of 3.6 cm and 180 kAsteady current with a 19.8 kA oscillation at 0.069 Hz, finding that thecell was nearly stable even when the salt layer thickness was reduced14% from its stable value (4.2 cm).

In an additional exemplary simulation, an ACD of 4.0 cm and 180 kAsteady current with a 19.8 kA oscillation at 0.0227 Hz was used. Thatfrequency drove the (2; 0) mode, whose frequency is predicted to be0.0229 Hz. Driving even that mode frustrated its ability to couple tothe (0; 1) mode and produce a circulation; the cell was nearly stable.

The claims are not intended to include, and should not be interpreted toinclude, means-plus- or step-plus-function limitations, unless such alimitation is explicitly recited in a given claim using the phrase(s)“means for” or “step for,” respectively.

In view of the described processes and compositions, hereinbelow aredescribed certain more particularly described aspects of the inventions.These particularly recited aspects should not, however, be interpretedto have any limiting effect on any different claims containing differentor more general teachings described herein, or that the “particular”aspects are somehow limited in some way other than the inherent meaningsof the language and formulas literally used therein.

Aspects:

Aspect 1: A system comprising: a) an electrolytic cell comprising: i) ananode; ii) a cathode; and iii) a molten electrolyte having apredetermined thickness; b) a direct current source that is inelectrical communication with the electrolytic cell and is configured toprovide a direct current (DC) having a predetermined amplitude and toinitiate an electrolysis reaction in the electrolytic cell; c) a devicecomprising an alternating current source; wherein the device is inelectrical communication with the electrolytic cell and is configured toprovide an alternating current (AC) to the electrolytic cell, whereinthe AC comprises an oscillatory current waveform defined by apredetermined amplitude, frequency, and/or phase; and wherein the deviceis in feedback loop communication with the electrolytic cell; andwherein the electrolytic cell exhibits substantially no change inoscillations present in the molten salt electrolyte over a predeterminedperiod of time when the AC is provided to the electrolytic cell.

Aspect 2: The system of Aspect 1, wherein the AC is further defined by anon-dimensional amplitude ratio β of the predetermined amplitude of theAC to the predetermined amplitude of the DC, wherein the β is fromgreater than 0 to about 1.

Aspect 3: The system of Aspect 1 or 2, wherein the predeterminedfrequency of the AC is from greater than 0 to about 120π rad/s.

Aspect 4: The system of any one of Aspects 1-3, wherein the electrolyticcell is an aluminum electrolysis cell.

Aspect 5: The system of any one of Aspects 1-4, wherein the moltenelectrolyte comprises cryolite.

Aspect 6: The system of any one of Aspects 1-5, wherein the anode and/orcathode comprises carbon.

Aspect 7: The system of any one of Aspects 1-6, wherein thepredetermined thickness of the molten electrolyte is equal to or lessthan about 4.5 cm.

Aspect 8: The system of any one of Aspects 1-7, wherein the directcurrent source is present in the device.

Aspect 9: The system of any one of Aspects 1-8, wherein the systemfurther comprises a unit configured to measure oscillations of themolten salt electrolyte as a function of the DC and AC applied to theelectrolytic cell; and wherein the unit is in feedback loopcommunication with the device and the electrolytic cell.

Aspect 10: The system of any one of Aspects 1-9, wherein the oscillatorycurrent waveform of AC comprises a sinusoidal, cosinusoidal, triangular,or square shape.

Aspect 11: The system of any one of Aspects 1-10, wherein thepredetermined amplitude, frequency, and/or phase of the AC is configuredto be adjusted in response to the feedback communication from theelectrolytic cell.

Aspect 12: A method comprising: a) providing an electrolytic cellcomprising: i) an anode; ii) a cathode; and iii) a molten electrolytehaving a predetermined thickness; b) applying a direct current (DC)having a predetermined amplitude to the electrolytic cell to initiate anelectrolysis reaction; c) applying an alternating current (AC)comprising an oscillatory current waveform defined by a predeterminedamplitude, frequency, and/or phase to the electrolytic cell, and d)measuring oscillations in the molten salt electrolyte as a function ofthe DC and AC applied to the electrolytic cell.

Aspect 13: The method of Aspect 12, wherein the electrolytic cellexhibits substantially no change in the oscillations present in themolten salt electrolyte over a predetermined period of time.

Aspect 14: The method of Aspect 12 or 13, wherein the AC is provided bya device comprising an alternating current source that is in electricalcommunication with the electrolytic cell.

Aspect 15: The method of Aspect 14, wherein the DC is provided by thedevice further comprising a direct current source.

Aspect 16: The method of any one of Aspects 12-14, wherein the DC isprovided by a separate direct current source.

Aspect 17: The method of any one of Aspects 12-16, wherein the measuringis performed with a unit that is in feedback loop communication with thedevice and the electrolytic cell.

Aspect 18: The method of any one of Aspects 14-17, wherein the device isin feedback loop communication with the electrolytic cell.

Aspect 19: The method of any one of Aspects 12-18, wherein theoscillatory current waveform comprises a sinusoidal, cosinusoidal,triangular, or square shape.

Aspect 20: The method of any one of Aspects 12-19, wherein the AC isfurther defined by a non-dimensional amplitude ratio β of thepredetermined amplitude of the AC to the predetermined amplitude of theDC, and wherein the β is from greater than 0 to about 1.

Aspect 21: The method of any one of Aspects 12-20, wherein thepredetermined frequency of the AC is from greater than 0 to about 120πrad/s.

Aspect 22: The method of any one of Aspects 20-21, further comprising astep of adjusting the predetermined amplitude, frequency, and/or phase,or β of AC based on the feedback communication from the electrolyticcell.

Aspect 23: The method of any one of Aspects 12-22, wherein theelectrolytic cell is an aluminum electrolysis cell.

Aspect 24: The method of any one of Aspects 12-23, wherein the moltenelectrolyte comprises cryolite.

Aspect 25: The method of any one of Aspects 12-24, wherein the anodeand/or cathode comprises carbon.

Aspect 26: The method of any one of Aspects 12-25, wherein thepredetermined thickness of the molten electrolyte is equal to or lessthan about 4.5 cm.

Aspect 27: A method comprising: applying an alternating current (AC)comprising an oscillatory current waveform to an electrolytic cellcomprising an electrolyte for a first predetermined time, whereinwaveform comprises an amplitude, frequency and/or phase that arepredetermined to stabilize the electrolytic cell such that substantiallyno change in a current oscillation is observed in the electrolyte duringelectrolysis.

Aspect 28: The method of Aspect 27, wherein the electrolyte is a moltensalt electrolyte.

Aspect 29: The method of Aspect 27 or 28, wherein a direct current (DC)is applied to the electrolytic cell for a second predetermined timeprior to applying the AC.

Aspect 30: The method of Aspect 29, wherein the direct current iscontinued to be applied for the first predetermined time simultaneouslywith applying the AC.

Aspect 31: The method of Aspect 27 or 28, wherein a direct current (DC)is applied to the electrolytic cell simultaneously with applying the ACfor the first predetermined time.

Aspect 32: The method of Aspect 29-30, wherein the second predeterminedtime is defined by a time period measured from applying the DC to anappearance of a resonant instability in the electrolytic cell.

Aspect 33: The method of any one of Aspects 27-32, wherein theelectrolytic cell further comprises an anode and a cathode.

Aspect 34: The method of Aspect 33, wherein the amplitude, frequencyand/or phase are predetermined such that an anode-to-cathode distance isreduced when it is compared to an anode-to-cathode distance of asubstantially identical reference electrolytic cell in the absence ofapplying an AC.

Aspect 35: The method of any one of Aspects 27-34, wherein theelectrolytic cell exhibits an increase in the energy efficiency when iscompared to the energy efficiency of a substantially identical referenceelectrolytic cell in the absence of applying the AC.

Aspect 36: The method of any one of Aspects 34-35, wherein theelectrolytic cell exhibits an increase in the energy efficiency that issubstantially proportional to a reduction in the anode-to-cathodedistance.

Aspect 37: The method of any one of Aspects 34-36, wherein theelectrolytic cell exhibits an increase in the energy efficiency of atleast about 5% when compared to the substantially identical referenceelectrolytic cell in the absence of applying an AC.

Aspect 38: The method of any one of Aspects 27-37, wherein the waveformis defined by a plurality of modes that together form a standing wave.

Aspect 39: The method of Aspect 38, wherein the plurality of modes ofthe waveform are configured to disrupt a formation and/or growth ofcirculating waves.

Aspect 40: The method of any one of Aspects 27-39, wherein the AC isprovided by a device comprising an alternating current source that is inelectrical communication with the electrolytic cell.

Aspect 41: The method of any one of Aspects 29-40, wherein the DC isprovided by the device further comprising a direct current source.

Aspect 42: The method of any one of Aspects 29-41, wherein the DC isprovided by a separate direct current source.

Aspect 43: The method of any one of Aspects 29-42, further comprisingmeasuring the current oscillations.

Aspect 44: The method of Aspect 43, wherein the measuring is performedby a controlling unit that is in a feedback loop communication with thedevice and the electrolytic cell such that it is configured to receivean input communication comprising a first data from the device and/orthe electrolytic cell and provide an output communication to the deviceand/or the electrolytic cell, wherein the output communication comprisesa second data adjusted for the first data.

Aspect 45: The method of any one of Aspects 43-44, wherein the device isin a feedback loop communication with the electrolytic cell.

Aspect 46: The method of any one of Aspects 27-45, wherein theoscillatory current waveform comprises a sinusoidal, cosinusoidal,triangular, or square shape.

Aspect 47: The method of any one of Aspects 29-46, wherein the AC isfurther defined by a non-dimensional amplitude ratio β of thepredetermined amplitude of the AC to the predetermined amplitude of theDC, and wherein the β is from greater than 0 to about 0.15.

Aspect 48: The method of any one of Aspects 27-47, wherein the frequencyof the AC is from about 0.01 Hz to about 0.5 Hz (about π/50 rad/s toabout n rad/s).

Aspect 49: The method of any one of Aspects 44-48, further comprising astep of adjusting the predetermined amplitude, frequency, and/or phase,or β of AC based on the feedback communication from the electrolyticcell.

Aspect 50: The method of any one of Aspects 27-49, wherein theelectrolytic cell is an aluminum electrolysis cell.

Aspect 51: The method of any one of Aspects 28-50, wherein the moltenelectrolyte comprises cryolite.

Aspect 52: The method of any one of Aspects 33-51, wherein the anodeand/or cathode comprises carbon.

Aspect 53: The method of any one of Aspects 27-52, wherein theelectrolyte comprises a thickness equal to or less than about 4.5 cm.

Aspect 54: The method of any one of Aspects 27-53, wherein the thicknessof the electrolyte is between about 3 cm to about 4.3 cm.

Aspect 55: A method comprising: a) providing a first data to acomputational processor, wherein the first data comprises at least oneof one or more of geometric parameters of an electrolytic cell, acathode-to-anode-distance of the electrolytic cell, a value of a directcurrent; an amplitude of a direct current, a thickness of a metal layer,material properties of a metal, material properties of an electrolyte,material properties of a cathode, material properties of an anode, orany combination thereof; b) analyzing the first data by thecomputational processor to provide a second data comprising parametersof an alternating current (AC) wherein the parameters comprise one ormore of a first amplitude, a first frequency, and/or a first phase of anoscillatory current form of the AC; and c) applying the AC having one ormore parameters present in the second data to the electrolytic cell tostabilize the electrolytic cell.

Aspect 56: The method of Aspect 55, further comprising: d) collecting athird data from the electrolytic cell and transferring the third data tothe computational processor to analyze the performance of theelectrolytic cell; e) analyzing the third data by the computationalprocessor to provide a fourth data comprising parameters of thealternating current (AC) wherein the parameters comprise one or more ofa second amplitude, a second frequency, and/or a second phase of anoscillatory current form of the AC; and f) applying the AC having one ormore parameters present in the fourth data to the electrolytic cell.

Aspect 57: The method of Aspect 56, comprising repeating steps of d)-f)until the electrolytic cell is stabilized such that substantially nochange in a current oscillation is observed in an electrolyte duringelectrolysis.

Aspect 58: A method for increasing energy efficiency in an electrolyticcell comprising: applying an alternating current (AC) comprising anoscillatory current waveform to the electrolytic cell comprising anelectrolyte for a first predetermined time, wherein waveform comprisesan amplitude, frequency and/or phase that are predetermined to stabilizethe electrolytic cell such that substantially no change in a currentoscillation is observed in the electrolyte during electrolysis; andwherein the energy efficiency is increased by at least about 5% whencompared to a substantially identical reference electrolytic cell in theabsence of applying an AC.

Aspect 59: A system comprising: a) an electrolytic cell comprising: i)an anode; iii) a cathode; and iii) an electrolyte having a predeterminedthickness; b) a direct current source that is in electricalcommunication with the electrolytic cell and is configured to provide adirect current (DC) having a predetermined amplitude and to initiate anelectrolysis reaction in the electrolytic cell; c) a device comprisingan alternating current source (AC); wherein the device is in electricalcommunication with the electrolytic cell and is configured to provide analternating current (AC) to the electrolytic cell, wherein the ACcomprises an oscillatory current waveform defined by a predeterminedamplitude, frequency, and/or phase; and wherein the device is infeedback loop communication with the electrolytic cell; and wherein theelectrolytic cell exhibits substantially no change in oscillationspresent in the molten salt electrolyte over a predetermined period oftime when the AC is provided to the electrolytic cell.

Aspect 60: The system of Aspect 59, wherein the AC is further defined bya non-dimensional amplitude ratio β of the predetermined amplitude ofthe AC to the predetermined amplitude of the DC, wherein the β is fromgreater than 0 to about 0.15.

Aspect 61: The system of Aspect 59 or 60, wherein the predeterminedfrequency of the AC is from about 0.01 Hz to about 0.5 Hz (about π/50rad/s to about π rad/s).

Aspect 62: The system of any one of Aspects 59-61, wherein theelectrolytic cell is an aluminum electrolysis cell.

Aspect 63: The system of any one of Aspects 59-62, wherein theelectrolyte is a molten electrolyte.

Aspect 64: The system of Aspect 63, wherein the molten electrolytecomprises cryolite.

Aspect 65: The system of any one of Aspects 59-64, wherein the anodeand/or cathode comprises carbon.

Aspect 66: The system of any one of Aspects 59-65, wherein thepredetermined thickness of the electrolyte is equal to or less thanabout 4.5 cm.

Aspect 67: The system of any one of Aspects 59-66, wherein the thicknessof the electrolyte is between about 3 cm to about 4.3 cm.

Aspect 68: The system of any one of Aspects 59-67, wherein theamplitude, frequency and/or phase are predetermined such that ananode-to-cathode distance is reduced when it is compared to ananode-to-cathode distance of a substantially identical referenceelectrolytic cell in the absence of providing the AC.

Aspect 69: The system of any one of Aspects 59-68, wherein theelectrolytic cell exhibits an increase in the energy efficiency when iscompared to the energy efficiency of a substantially identical referenceelectrolytic cell in the absence of providing the AC.

Aspect 70: The system of any one of Aspects 59-69, wherein theelectrolytic cell exhibits an increase in the energy efficiency that issubstantially proportional to a reduction in the anode-to-cathodedistance.

Aspect 71: The system of any one of Aspects 59-70, wherein theelectrolytic cell exhibits an increase in the energy efficiency of atleast about 5% when is compared to the substantially identical referenceelectrolytic cell in the absence of providing the AC.

Aspect 72: The system of any one of Aspects 61-71, wherein the waveformis defined by a plurality of modes that together form a standing wave.

Aspect 73: The system of Aspect 72, wherein the plurality of modes ofthe waveform are configured to disrupt a formation and/or growth ofcirculating waves.

Aspect 74: The system of any one of Aspects 59-73, wherein the directcurrent source is present in the device.

Aspect 75: The system of any one of Aspects 59-74, wherein the systemfurther comprises a controlling unit configured to measure oscillationsof the molten salt electrolyte as a function of the DC and AC applied tothe electrolytic cell; and wherein the controlling unit is in a feedbackloop communication with the device and the electrolytic cell.

Aspect 76: The system of any one of Aspects 59-75, wherein theoscillatory current waveform of AC comprises a sinusoidal, cosinusoidal,triangular, or square shape.

Aspect 77: The system of Aspect 59-76, wherein the predeterminedamplitude, frequency, and/or phase of the AC is configured to beadjusted in response to the feedback communication from the electrolyticcell.

1. A method comprising: applying an alternating current (AC) comprisingan oscillatory current waveform to an electrolytic cell comprising anelectrolyte, an anode, and a cathode for a first predetermined time,wherein waveform comprises an amplitude, frequency and/or phase that arepredetermined to stabilize the electrolytic cell such that substantiallyno change in a current oscillation is observed in the electrolyte duringelectrolysis.
 2. (canceled)
 3. The method of claim 1, wherein a directcurrent (DC) is applied to the electrolytic cell for a secondpredetermined time prior to applying the AC, wherein the secondpredetermined time is defined by a time period measured from applyingthe DC to an appearance of a resonant instability in the electrolyticcell.
 4. The method of claim 3, wherein the direct current is continuedto be applied for the first predetermined time simultaneously withapplying the AC.
 5. The method of claim 1, wherein a direct current (DC)is applied to the electrolytic cell simultaneously with applying the ACfor the first predetermined time.
 6. (canceled)
 7. (canceled)
 8. Themethod of claim 1, wherein the amplitude, frequency and/or phase arepredetermined such that an anode-to-cathode distance is reduced when itis compared to an anode-to-cathode distance of a substantially identicalreference electrolytic cell in the absence of applying an AC, andwherein the electrolytic cell exhibits an increase in the energyefficiency that is substantially proportional to a reduction in theanode-to-cathode distance.
 9. (canceled)
 10. (canceled)
 11. (canceled)12. The method of claim 1, wherein the waveform is defined by aplurality of modes that together form a standing wave, and wherein theplurality of modes of the waveform are configured to disrupt formationand/or growth of circulating waves.
 13. (canceled)
 14. The method ofclaim 1, wherein the AC is provided by a device comprising analternating current source that is in electrical communication with theelectrolytic cell.
 15. The method of claim 14, wherein, the devicefurther comprises a direct current source configured to provide the DCor wherein the DC is provided by a separate direct current source. 16.(canceled)
 17. The method of claim 1, further comprising measuring thecurrent oscillations, wherein the measuring the current oscillations isperformed by a controlling unit that is in a feedback loop communicationwith the device and the electrolytic cell such that it is configured toreceive an input communication comprising a first data from the deviceand/or the electrolytic cell and provide an output communication to thedevice and/or the electrolytic cell, wherein the output communicationcomprises a second data adjusted for the first data.
 18. (canceled) 19.(canceled)
 20. (canceled)
 21. The method of claim 3, wherein the AC isfurther defined by a non-dimensional amplitude ratio β of thepredetermined amplitude of the AC to the predetermined amplitude of theDC, and wherein the β is from greater than 0 to about 0.15, and/orwherein the frequency of the AC is from about 0.01 Hz to about 0.5 Hz(about π/50 rad/s to about π rad/s).
 22. (canceled)
 23. The method ofclaim 17, further comprising a step of adjusting the predeterminedamplitude, frequency, and/or phase, or β of AC based on the feedbackcommunication from the electrolytic cell.
 24. The method of claim 1,wherein the electrolytic cell is an aluminum electrolysis cell andwherein the electrolyte is a molten salt electrolyte.
 25. (canceled) 26.(canceled)
 27. The method of claim 1, wherein the electrolyte comprisesa thickness equal to or less than about 4.5 cm.
 28. (canceled)
 29. Amethod comprising: a) providing a first data to a computationalprocessor, wherein the first data comprises at least one of one or moreof geometric parameters of an electrolytic cell, acathode-to-anode-distance of the electrolytic cell, a value of a directcurrent; an amplitude of a direct current, a thickness of a metal layer,material properties of a metal, material properties of an electrolyte,material properties of a cathode, material properties of an anode, orany combination thereof; b) analyzing the first data by thecomputational processor to provide a second data comprising parametersof an alternating current (AC) wherein the parameters comprise one ormore of a first amplitude, a first frequency, and/or a first phase of anoscillatory current form of the AC; c) applying the AC having one ormore parameters present in the second data to the electrolytic cell tostabilize the electrolytic cell.
 30. The method of claim 29, furthercomprising: d) collecting a third data from the electrolytic cell andtransferring the third data to the computational processor to analyzethe performance of the electrolytic cell; e) analyzing the third data bythe computational processor to provide a fourth data comprisingparameters of the alternating current (AC) wherein the parameterscomprise one or more of a second amplitude, a second frequency, and/or asecond phase of an oscillatory current form of the AC; and f) applyingthe AC having one or more parameters present in the fourth data to theelectrolytic cell; and optionally comprising repeating steps of d)-f)until the electrolytic cell is stabilized such that substantially nochange in a current oscillation is observed in an electrolyte duringelectrolysis.
 31. (canceled)
 32. (canceled)
 33. A system comprising: a)an electrolytic cell comprising: i) an anode; ii) a cathode; and iii) amolten salt electrolyte having a predetermined thickness; b) a directcurrent source that is in electrical communication with the electrolyticcell and is configured to provide a direct current (DC) having apredetermined amplitude and to initiate an electrolysis reaction in theelectrolytic cell; c) a device comprising an alternating current source(AC); wherein the device is in electrical communication with theelectrolytic cell and is configured to provide an alternating current(AC) to the electrolytic cell, wherein the AC comprises an oscillatorycurrent waveform defined by a predetermined amplitude, frequency, and/orphase; and wherein the device is in feedback loop communication with theelectrolytic cell; wherein the AC is further defined by anon-dimensional amplitude ratio β of the predetermined amplitude of theAC to the predetermined amplitude of the DC, wherein the β is fromgreater than 0 to about 0.15; wherein the predetermined frequency of theAC is from about 0.01 Hz to about 0.5 Hz (about π/50 rad/s to about πrad/s); and wherein the electrolytic cell exhibits substantially nochange in oscillations present in the molten salt electrolyte over apredetermined period of time when the AC is provided to the electrolyticcell.
 34. (canceled)
 35. (canceled)
 36. The system of claim 33, whereinthe electrolytic cell is an aluminum electrolysis cell.
 37. (canceled)38. (canceled)
 39. (canceled)
 40. The system of claim 33, wherein thepredetermined thickness of the electrolyte is equal to or less thanabout 4.5 cm.
 41. (canceled)
 42. The system of claim 33, wherein theamplitude, frequency and/or phase are predetermined such that ananode-to-cathode distance is reduced when it is compared to ananode-to-cathode distance of a substantially identical referenceelectrolytic cell in the absence of providing the AC, and wherein theelectrolytic cell exhibits an increase in the energy efficiency that issubstantially proportional to a reduction in the anode-to-cathodedistance.
 43. (canceled)
 44. (canceled)
 45. (canceled)
 46. (canceled)47. (canceled)
 48. (canceled)
 49. The system of claim 33, wherein thesystem further comprises a controlling unit configured to measureoscillations of the molten salt electrolyte as a function of the DC andAC applied to the electrolytic cell; and wherein the controlling unit isin a feedback loop communication with the device and the electrolyticcell.
 50. (canceled)
 51. (canceled)